A254327 -id:A254327 - cp's OEIS frontend

A251866 Decimal expansion of gamma_1(1/5), the first generalized Stieltjes constant at 1/5 (negated).

8, 0, 3, 0, 2, 0, 5, 5, 1, 1, 0, 3, 5, 9, 7, 6, 8, 8, 7, 6, 2, 7, 8, 9, 1, 3, 4, 6, 6, 5, 1, 0, 3, 4, 8, 5, 3, 9, 9, 8, 6, 3, 8, 6, 9, 5, 2, 7, 4, 3, 7, 6, 8, 1, 0, 5, 4, 5, 3, 1, 6, 6, 6, 6, 1, 7, 7, 5, 3, 8, 1, 6, 4, 0, 6, 8, 9, 8, 5, 6, 2, 5, 1, 7, 7, 5, 0, 8, 0, 6, 2, 5, 4, 9, 9, 4, 3, 0, 8, 4, 4, 1, 5, 4, 6

Offset: 1

Jean-François Alcover, Feb 16 2015

			-8.03020551103597688762789134665103485399863869527437681...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Iaroslav V. Blagouchine, A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments, arXiv:1401.3724 [math.NT], 2015.
Iaroslav V. Blagouchine, A theorem ... (same title), Journal of Number Theory Volume 148, March 2015, Pages 537-592.
Iaroslav V. Blagouchine, Rediscovery of Malmsten’s integrals, their evaluation by contour integration methods and some related results, The Ramanujan Journal October 2014, Volume 35, Issue 1, pp 21-110.
Iaroslav V. Blagouchine, Rediscovery of Malmsten’s integrals: Full PDF text.
Eric Weisstein's World of Mathematics, Hurwitz Zeta Function.
Eric Weisstein's World of Mathematics, Stieltjes Constants.
Wikipedia, Stieltjes constants

Cf. A001620 (gamma), A082633 (gamma_1), A254327 (gamma_1(1/2)), A254331 (gamma_1(1/3)), A254345 (gamma_1(2/3)), A254347 (gamma_1(1/4)), A254348 (gamma_1(3/4)), A254349 (gamma_1(1/6)), A254350 (gamma_1(5/6)).

gamma1[1/5] = StieltjesGamma[1] + (1/2)*Sqrt[5]*(Derivative[2, 0][Zeta][0, 1/5] + Derivative[2, 0][Zeta][0, 4/5]) + (1/2)*(Pi*Sqrt[10 + 2*Sqrt[5]])*LogGamma[1/5] + (1/2)*(Pi*Sqrt[10 - 2*Sqrt[5]])*LogGamma[2/5] + ((1/2)*Sqrt[5]*Log[2] - (1/2)* Sqrt[5] *Log[1 + Sqrt[5]] - (1/10)*Pi*Sqrt[25 + 10*Sqrt[5]] -(5*Log[5])/4) *EulerGamma - (1/2)*Sqrt[5]*(Log[2] + Log[5] + Log[Pi] + (1/10)*Sqrt[25 - 10*Sqrt[5]] *Pi)*Log[1 + Sqrt[5]] + (1/2)*Sqrt[5]*Log[2]^2 + (1/8)*(Sqrt[5]*(1 - Sqrt[5]))*Log[5]^2 + ((3*Sqrt[5]) /4) *Log[2]*Log[5] + (Sqrt[5]/2)*Log[2]*Log[Pi] + (Sqrt[5]/4) *Log[5] *Log[Pi] - ((Pi*(2*Sqrt[25 + 10*Sqrt[5]] + 5*Sqrt[25 + 2*Sqrt[5]]))/20)*Log[2] -((Pi*(4*Sqrt[25 + 10*Sqrt[5]] - 5*Sqrt[5 + 2*Sqrt[5]]))/40)*Log[5] - ((Pi*(5*Sqrt[5 + 2*Sqrt[5]] + Sqrt[25 + 10*Sqrt[5]]))/10)*Log[Pi] // Re; RealDigits[gamma1[1/5], 10, 105] // First
(* or, from version 7 up: *) RealDigits[StieltjesGamma[1, 1/5], 10, 105] // First

A254349 Decimal expansion of gamma_1(1/6), the first generalized Stieltjes constant at 1/6 (negated).

Original entry on oeis.org

1, 0, 7, 4, 2, 5, 8, 2, 5, 2, 9, 5, 4, 7, 8, 9, 2, 2, 5, 8, 9, 4, 1, 1, 9, 6, 7, 7, 6, 2, 4, 3, 6, 6, 8, 3, 0, 1, 6, 3, 0, 4, 2, 6, 1, 6, 3, 6, 0, 6, 7, 5, 3, 7, 9, 5, 1, 6, 4, 5, 8, 4, 3, 9, 6, 8, 7, 3, 7, 2, 8, 3, 6, 6, 9, 6, 1, 0, 0, 9, 2, 3, 3, 8, 9, 8, 9, 6, 9, 5, 6, 3, 1, 9, 9, 6, 7, 3, 8, 6, 9, 6

Offset: 2

Jean-François Alcover, Jan 29 2015

			-10.742582529547892258941196776243668301630426163606753795...

G. C. Greubel, Table of n, a(n) for n = 2..5002
Iaroslav V. Blagouchine, A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments, arXiv:1401.3724 [math.NT], 2015.
Iaroslav V. Blagouchine, A theorem ... (same title), Journal of Number Theory Volume 148, March 2015, Pages 537-592.
Iaroslav V. Blagouchine, Rediscovery of Malmsten’s integrals, their evaluation by contour integration methods and some related results, The Ramanujan Journal October 2014, Volume 35, Issue 1, pp 21-110.
Iaroslav V. Blagouchine, Rediscovery of Malmsten’s integrals: Full PDF text.
Eric Weisstein's World of Mathematics, Hurwitz Zeta Function;
Eric Weisstein's World of Mathematics, Stieltjes Constants.
Wikipedia, Stieltjes constants

Cf. A001620 (gamma), A082633 (gamma_1), A254327 (gamma_1(1/2)), A254331 (gamma_1(1/3)), A254345 (gamma_1(2/3)), A254347 (gamma_1(1/4)), A254348 (gamma_1(3/4)), A254350 (gamma_1(5/6)), A251866 (gamma_1(1/5)), A255188 (gamma_1(1/8)), A255189 (gamma_1(1/12)).

evalf(int((6*(-2*arctan(6*x) + 6*x*log(1/36 + x^2)))/((-1 + exp(2*Pi*x))*(1 + 36*x^2)), x = 0..infinity) - (3 + log(6)/2)*log(6), 120); # Vaclav Kotesovec, Jan 29 2015

gamma1[1/6] = (1/2)*((-Log[6])*Log[24] - EulerGamma*Log[432] - 2*Log[2]*Log[2*Pi^2] + Log[(2*Pi)/Sqrt[3]]*Log[144*Pi^2] + Log[Pi]*Log[4/Gamma[1/6]^2] - 2*Log[12]*Log[Gamma[1/6]] - 2*Log[12*Pi]*Log[Gamma[5/6]] - Sqrt[3]*Pi*(EulerGamma + Log[(12*2^(2/3)*Pi^(3/2)*Gamma[5/6])/Gamma[1/6]^2]) + 2*StieltjesGamma[1] + Derivative[2, 0][Zeta][0, 1/6] - Derivative[2, 0][Zeta][0, 1/3] - 2*Derivative[2, 0][Zeta][0, 1/2] - Derivative[2, 0][Zeta][0, 2/3] + Derivative[2, 0][Zeta][0, 5/6]) // Re; RealDigits[gamma1[1/6], 10, 102] // First
(* Or, from Mma version 7 up: *) RealDigits[StieltjesGamma[1, 1/6], 10, 102] // First

Equals integral_[0..infinity] (6*(-2*arctan(6*x) + 6*x*log(1/36 + x^2)))/((-1 + exp(2*Pi*x))*(1 + 36*x^2)) dx - (3 + log(6)/2)*log(6).

A254331 Decimal expansion of gamma_1(1/3), the first generalized Stieltjes constant at 1/3 (negated).

Original entry on oeis.org

3, 2, 5, 9, 5, 5, 7, 5, 1, 5, 9, 1, 7, 9, 1, 0, 1, 9, 5, 2, 5, 0, 8, 7, 4, 5, 8, 2, 6, 7, 6, 5, 5, 9, 2, 5, 7, 9, 7, 6, 4, 7, 2, 2, 0, 4, 3, 9, 9, 4, 3, 0, 0, 4, 8, 1, 1, 7, 9, 7, 4, 8, 6, 7, 3, 8, 9, 7, 9, 3, 7, 0, 1, 4, 9, 5, 4, 6, 8, 7, 9, 2, 4, 7, 8, 9, 6, 5, 2, 5, 8, 8, 2, 0, 0, 8, 6, 7, 3, 8, 0, 4, 3, 2

Offset: 1

Jean-François Alcover, Jan 28 2015

			-3.25955751591791019525087458267655925797647220439943...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Iaroslav V. Blagouchine, A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments, arXiv:1401.3724 [math.NT], 2015.
Iaroslav V. Blagouchine, A theorem ... (same title), Journal of Number Theory Volume 148, March 2015, Pages 537-592.
Iaroslav V. Blagouchine, Rediscovery of Malmsten’s integrals, their evaluation by contour integration methods and some related results, The Ramanujan Journal October 2014, Volume 35, Issue 1, pp 21-110.
Iaroslav V. Blagouchine, Rediscovery of Malmsten’s integrals: Full PDF text.
Eric Weisstein's World of Mathematics, Hurwitz Zeta Function.
Eric Weisstein's World of Mathematics, Stieltjes Constants.
Wikipedia, Stieltjes constants

Cf. A001620 (gamma), A082633 (gamma_1), A254327 (gamma_1(1/2)), A254345 (gamma_1(2/3)), A254347 (gamma_1(1/4)), A254348 (gamma_1(3/4)), A254349 (gamma_1(1/6)), A254350 (gamma_1(5/6)), A251866 (gamma_1(1/5)), A255188 (gamma_1(1/8)), A255189 (gamma_1(1/12)).

evalf(int((3*(-2*arctan(3*x)+3*x*log(1/9+x^2)))/((-1+exp(2*Pi*x))*(9*x^2+1)), x = 0..infinity)-3*log(3)*(1/2)-(1/2)*log(3)^2, 120); # Vaclav Kotesovec, Jan 29 2015

gamma1[1/3] = (1/6)*((-Sqrt[3])*Pi*(EulerGamma + Log[(24*Pi^(5/2))/Gamma[1/6]^3]) - 3*(Log[3]^2 + EulerGamma*Log[27] + 2*Log[3]*Log[2*Pi] + 2*Log[2*Pi]^2 + Log[3/(4*Pi^2)]*Log[6*Pi] - 2*StieltjesGamma[1] + Derivative[2, 0][Zeta][0, 1/3] + Derivative[2, 0][Zeta][0, 2/3])) // Re; RealDigits[gamma1[1/3], 10, 104] // First
(* Or, from Mma version 7 up: *) RealDigits[StieltjesGamma[1, 1/3], 10, 104] // First

Equals integral_[0..infinity] (3*(-2*arctan(3*x) + 3*x*log(1/9 + x^2)))/((-1 + exp(2*Pi*x))*(1 + 9*x^2)) dx - 3*log(3)/2 - log(3)^2/2.

A254345 Decimal expansion of gamma_1(2/3), the first generalized Stieltjes constant at 2/3 (negated).

Original entry on oeis.org

5, 9, 8, 9, 0, 6, 2, 8, 4, 2, 8, 5, 9, 8, 9, 2, 9, 2, 5, 6, 7, 8, 7, 6, 0, 2, 1, 2, 6, 9, 2, 5, 0, 2, 5, 6, 6, 6, 3, 9, 1, 3, 4, 0, 7, 8, 1, 7, 5, 7, 1, 4, 9, 1, 5, 8, 6, 5, 0, 1, 5, 6, 9, 7, 1, 8, 7, 2, 0, 7, 6, 5, 0, 2, 5, 5, 0, 4, 7, 8, 6, 7, 7, 3, 3, 4, 2, 4, 7, 9, 8, 8, 1, 7, 2, 9, 0, 7, 1, 1, 1, 5, 2, 9, 8

Offset: 0

Jean-François Alcover, Jan 29 2015

			-0.5989062842859892925678760212692502566639134078175714915865...

G. C. Greubel, Table of n, a(n) for n = 0..5000
Iaroslav V. Blagouchine, A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments, arXiv:1401.3724 [math.NT], 2015.
Iaroslav V. Blagouchine, A theorem ... (same title), Journal of Number Theory Volume 148, March 2015, Pages 537-592.
Iaroslav V. Blagouchine, Rediscovery of Malmsten’s integrals, their evaluation by contour integration methods and some related results, The Ramanujan Journal October 2014, Volume 35, Issue 1, pp 21-110.
Iaroslav V. Blagouchine, Rediscovery of Malmsten’s integrals: Full PDF text.
Eric Weisstein's World of Mathematics, Hurwitz Zeta Function;
Eric Weisstein's World of Mathematics, Stieltjes Constants.
Wikipedia, Stieltjes constants

Cf. A001620 (gamma), A082633 (gamma_1), A254327 (gamma_1(1/2)), A254331 (gamma_1(1/3)), A254347 (gamma_1(1/4)), A254348 (gamma_1(3/4)), A254349 (gamma_1(1/6)), A254350 (gamma_1(5/6)), A251866 (gamma_1(1/5)), A255188 (gamma_1(1/8)), A255189 (gamma_1(1/12)).

evalf(int((-12*arctan(3*x*(1/2))+9*x*log(4/9+x^2))/((-1+exp(2*Pi*x))*(9*x^2+4)), x = 0..infinity) - (3/4+(1/2)*log(3/2))*log(3/2), 120); # Vaclav Kotesovec, Jan 29 2015

gamma1[2/3] = (1/12)*(Sqrt[3]*Pi*(2*EulerGamma + Log[(576*Pi^5)/Gamma[1/6]^6]) - 6*(EulerGamma*Log[27] + Log[3]*Log[18*Pi] - 2*StieltjesGamma[1] + Derivative[2, 0][Zeta][0, 1/3] + Derivative[2, 0][Zeta][0, 2/3])) // Re; RealDigits[gamma1[2/3], 10, 105] // First
(* Or, from Mma version 7 up: *) RealDigits[StieltjesGamma[1, 2/3], 10, 105] // First

Equals integral_[0..infinity] (-12*arctan(3*x/2) + 9*x*log(4/9 + x^2))/((-1 + exp(2*Pi*x))*(4 + 9*x^2)) dx - (3/4 + (1/2)*log(3/2))*log(3/2).

A254347 Decimal expansion of gamma_1(1/4), the first generalized Stieltjes constant at 1/4 (negated).

Original entry on oeis.org

5, 5, 1, 8, 0, 7, 6, 3, 5, 0, 1, 9, 9, 4, 0, 3, 7, 5, 2, 6, 9, 4, 0, 1, 1, 0, 4, 4, 7, 7, 6, 6, 5, 5, 4, 0, 7, 1, 0, 7, 9, 4, 4, 6, 0, 3, 1, 8, 5, 7, 4, 3, 4, 6, 3, 6, 1, 4, 2, 9, 4, 5, 2, 4, 8, 6, 0, 2, 1, 9, 3, 0, 7, 7, 8, 5, 0, 7, 0, 3, 8, 7, 0, 6, 9, 7, 0, 8, 4, 1, 9, 4, 9, 9, 0, 3, 7, 4, 8, 0, 1, 5, 5

Offset: 1

Jean-François Alcover, Jan 29 2015

			-5.5180763501994037526940110447766554071079446031857434636...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Iaroslav V. Blagouchine, A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments, arXiv:1401.3724 [math.NT], 2015.
Iaroslav V. Blagouchine, A theorem ... (same title), Journal of Number Theory Volume 148, March 2015, Pages 537-592.
Iaroslav V. Blagouchine, Rediscovery of Malmsten’s integrals, their evaluation by contour integration methods and some related results, The Ramanujan Journal October 2014, Volume 35, Issue 1, pp 21-110.
Iaroslav V. Blagouchine, Rediscovery of Malmsten’s integrals: Full PDF text.
Eric Weisstein's World of Mathematics, Hurwitz Zeta Function.
Eric Weisstein's World of Mathematics, Stieltjes Constants.
Wikipedia, Stieltjes constants

Cf. A001620 (gamma), A082633 (gamma_1), A254327 (gamma_1(1/2)), A254331 (gamma_1(1/3)), A254345 (gamma_1(2/3)), A254348 (gamma_1(3/4)), A254349 (gamma_1(1/6)), A254350 (gamma_1(5/6)), A251866 (gamma_1(1/5)), A255188 (gamma_1(1/8)), A255189 (gamma_1(1/12)).

evalf(int((4*(-2*arctan(4*x)+4*x*log(1/16+x^2)))/((-1+exp(2*Pi*x))*(16*x^2+1)), x = 0..infinity) - (2+(1/2)*log(4))*log(4), 120); # Vaclav Kotesovec, Jan 29 2015

gamma1[1/4] = -1/2*Log[4]^2 - 1/2*EulerGamma*(Pi + Log[64]) - Log[4]*Log[2*Pi] - Log[2*Pi]^2 + Log[Pi]*Log[8*Pi] - 1/2*Pi*Log[8*Pi*Gamma[3/4]^2/Gamma[1/4]^2] + StieltjesGamma[1] - Derivative[2, 0][Zeta][0, 1/2] // Re; RealDigits[gamma1[1/4], 10, 103] // First
(* Or, from Mma version 7 up: *) RealDigits[StieltjesGamma[1, 1/4], 10, 103] // First

Equals integral_[0..infinity] (4*(-2*arctan(4*x) + 4*x*log(1/16 + x^2)))/((-1 + exp(2*Pi*x))*(1 + 16*x^2)) dx - (2 + log(4)/2)*log(4).

A254348 Decimal expansion of gamma_1(3/4), the first generalized Stieltjes constant at 3/4 (negated).

Original entry on oeis.org

3, 9, 1, 2, 9, 8, 9, 0, 2, 4, 0, 4, 5, 4, 9, 7, 7, 4, 2, 3, 9, 8, 7, 4, 1, 9, 2, 1, 8, 9, 2, 9, 6, 3, 7, 1, 4, 5, 0, 3, 8, 9, 7, 3, 1, 9, 6, 7, 1, 4, 0, 7, 6, 6, 2, 7, 7, 3, 0, 7, 1, 0, 8, 6, 9, 7, 1, 7, 9, 3, 9, 5, 0, 6, 0, 4, 7, 1, 3, 3, 2, 6, 4, 3, 2, 7, 8, 2, 7, 5, 6, 2, 2, 1, 9, 7, 5, 8, 8, 1, 4, 7, 8

Offset: 0

Jean-François Alcover, Jan 29 2015

			-0.39129890240454977423987419218929637145038973196714...

G. C. Greubel, Table of n, a(n) for n = 0..5000
Iaroslav V. Blagouchine, A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments, arXiv:1401.3724 [math.NT], 2015.
Iaroslav V. Blagouchine, A theorem ... (same title), Journal of Number Theory Volume 148, March 2015, Pages 537-592.
Iaroslav V. Blagouchine, Rediscovery of Malmsten’s integrals, their evaluation by contour integration methods and some related results, The Ramanujan Journal October 2014, Volume 35, Issue 1, pp 21-110.
Iaroslav V. Blagouchine, Rediscovery of Malmsten’s integrals: Full PDF text.
Eric Weisstein's World of Mathematics, Hurwitz Zeta Function.
Eric Weisstein's World of Mathematics, Stieltjes Constants.
Wikipedia, Stieltjes constants

Cf. A001620 (gamma), A082633 (gamma_1), A254327 (gamma_1(1/2)), A254331 (gamma_1(1/3)), A254345 (gamma_1(2/3)), A254347 (gamma_1(1/4)), A254349 (gamma_1(1/6)), A254350 (gamma_1(5/6)), A251866 (gamma_1(1/5)), A255188 (gamma_1(1/8)), A255189 (gamma_1(1/12)).

evalf(int((4*(-6*arctan(4*x*(1/3))+4*x*log(9/16+x^2)))/((-1+exp(2*Pi*x))*(16*x^2+9)), x = 0..infinity) - (2/3+(1/2)*log(4/3))*log(4/3), 120); # Vaclav Kotesovec, Jan 29 2015

gamma1[3/4] = (1/2)*(-Log[4]^2 + EulerGamma*(Pi - 2*Log[8]) - 2*Log[4]*Log[2*Pi] + Pi*Log[(8*Pi*Gamma[3/4]^2)/Gamma[1/4]^2] - 2*(Log[2*Pi]^2 - Log[Pi]*Log[8*Pi] - StieltjesGamma[1] + Derivative[2, 0][Zeta][0, 1/2])) // Re; RealDigits[gamma1[3/4], 10, 103] // First
(* Or, from Mma version 7 up: *) RealDigits[StieltjesGamma[1, 3/4], 10, 103] // First

Equals integral_[0..infinity] (4*(-6*arctan(4*x/3) + 4*x*log(9/16 + x^2)))/((-1 + exp(2*Pi*x))*(9 + 16*x^2)) dx -(2/3 + (1/2)*log(4/3))*log(4/3).

A254350 Decimal expansion of gamma_1(5/6), the first generalized Stieltjes constant at 5/6 (negated).

Original entry on oeis.org

2, 4, 6, 1, 6, 9, 0, 0, 3, 8, 1, 1, 3, 9, 0, 7, 3, 3, 1, 4, 8, 4, 9, 1, 7, 1, 5, 3, 2, 7, 4, 9, 0, 6, 9, 5, 7, 7, 0, 8, 6, 9, 0, 9, 0, 1, 2, 8, 4, 4, 2, 3, 2, 9, 7, 9, 6, 4, 3, 3, 2, 6, 6, 5, 0, 2, 0, 4, 3, 1, 3, 5, 5, 1, 7, 4, 5, 1, 0, 4, 9, 8, 1, 9, 1, 3, 4, 1, 5, 5, 5, 8, 6, 5, 7, 0, 6, 6, 1, 6, 8, 5, 5, 4

Offset: 0

Jean-François Alcover, Jan 29 2015

			-0.24616900381139073314849171532749069577086909012844...

G. C. Greubel, Table of n, a(n) for n = 0..5000
Iaroslav V. Blagouchine, A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments, arXiv:1401.3724 [math.NT], 2015.
Iaroslav V. Blagouchine, A theorem ... (same title), Journal of Number Theory Volume 148, March 2015, Pages 537-592.
Iaroslav V. Blagouchine, Rediscovery of Malmsten’s integrals, their evaluation by contour integration methods and some related results, The Ramanujan Journal October 2014, Volume 35, Issue 1, pp 21-110.
Iaroslav V. Blagouchine, Rediscovery of Malmsten’s integrals: Full PDF text.
Eric Weisstein's World of Mathematics, Hurwitz Zeta Function.
Eric Weisstein's World of Mathematics, Stieltjes Constants.
Wikipedia, Stieltjes constants

Cf. A001620 (gamma), A082633 (gamma_1), A254327 (gamma_1(1/2)), A254331 (gamma_1(1/3)), A254345 (gamma_1(2/3)), A254347 (gamma_1(1/4)), A254348 (gamma_1(3/4)), A254349 (gamma_1(1/6)), A251866 (gamma_1(1/5)), A255188 (gamma_1(1/8)), A255189 (gamma_1(1/12)).

gamma1[5/6] = (1/2)*((-Log[6])*Log[24] - EulerGamma*Log[432] - 2*Log[2]*Log[2*Pi^2] + Log[(2*Pi)/Sqrt[3]]*Log[144*Pi^2] + Log[Pi]*Log[4/Gamma[1/6]^2] - *Log[12] * Log[Gamma[1/6]] - 2*Log[12*Pi]*Log[Gamma[5/6]] + Sqrt[3]*Pi*(EulerGamma + Log[(12*2^(2/3)*Pi^(3/2)*Gamma[5/6])/Gamma[1/6]^2]) + 2*StieltjesGamma[1] + Derivative[2, 0][Zeta][0, 1/6] - Derivative[2, 0][Zeta][0, 1/3] -
      2*Derivative[2, 0][Zeta][0, 1/2] - Derivative[2, 0][Zeta][0, 2/3] + Derivative[2, 0][Zeta][0, 5/6]) // Re; RealDigits[gamma1[5/6], 10, 104] // First
(* Or, from Mma version 7 up: *) RealDigits[StieltjesGamma[1, 5/6], 10, 104] // First

Equals integral_[0..infinity] (6*(-10*arctan((6*x)/5) + 6*x*log(25/36 + x^2)))/((-1 + e^(2*Pi*x))*(25 + 36*x^2)) dx -(3/5 + (1/2)*log(6/5))*log(6/5).

A255188 Decimal expansion of gamma_1(1/8), the first generalized Stieltjes constant at 1/8 (negated).

Original entry on oeis.org

1, 6, 6, 4, 1, 7, 1, 9, 7, 6, 3, 6, 0, 9, 3, 1, 5, 6, 6, 2, 8, 4, 1, 9, 2, 6, 2, 3, 0, 3, 7, 3, 9, 4, 4, 9, 2, 8, 5, 1, 3, 2, 6, 6, 0, 6, 5, 4, 7, 4, 4, 5, 5, 2, 9, 4, 2, 9, 3, 7, 9, 2, 5, 1, 8, 2, 2, 9, 3, 6, 5, 2, 4, 9, 2, 2, 3, 8, 1, 5, 7, 1, 5, 4, 1, 4, 5, 7, 7, 1, 7, 3, 9, 1, 9, 0, 6, 3, 2, 0, 7, 5, 6, 8

Offset: 2

Jean-François Alcover, Feb 16 2015

			-16.641719763609315662841926230373944928513266065474455...

G. C. Greubel, Table of n, a(n) for n = 2..5002
Iaroslav V. Blagouchine, A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments, arXiv:1401.3724 [math.NT], 2015.
Iaroslav V. Blagouchine, A theorem ... (same title), Journal of Number Theory Volume 148, March 2015, Pages 537-592.
Iaroslav V. Blagouchine, Rediscovery of Malmsten’s integrals, their evaluation by contour integration methods and some related results, The Ramanujan Journal October 2014, Volume 35, Issue 1, pp 21-110.
Iaroslav V. Blagouchine, Rediscovery of Malmsten’s integrals: Full PDF text.
Eric Weisstein's World of Mathematics, Hurwitz Zeta Function.
Eric Weisstein's World of Mathematics, Stieltjes Constants.
Wikipedia, Stieltjes constants

Cf. A001620 (gamma), A082633 (gamma_1), A254327 (gamma_1(1/2)), A254331 (gamma_1(1/3)), A254345 (gamma_1(2/3)), A254347 (gamma_1(1/4)), A254348 (gamma_1(3/4)), A254349 (gamma_1(1/6)), A254350 (gamma_1(5/6)), A251866 (gamma_1(1/5)).

gamma1[1/8] = StieltjesGamma[1] + Sqrt[2]*(Derivative[2, 0][Zeta][0, 1/8] + Derivative[2, 0][Zeta][0, 7/8]) + 2*Pi*Sqrt[2]*LogGamma[1/8] - Pi*Sqrt[2]*(1 - Sqrt[2]) *LogGamma[1/4] - ((1 + Sqrt[2])*(Pi/2) + 4*Log[2] + Sqrt[2]*Log[1 + Sqrt[2]])* EulerGamma - (1/Sqrt[2])*(Pi + 8*Log[2] + 2*Log[Pi])*Log[1 + Sqrt[2]] - 7*((4 - Sqrt[2] )/4)*Log[2]^2 + (1/Sqrt[2])*Log[2]*Log[Pi] - Pi*((10 + 11*Sqrt[2])/4)*Log[2] - Pi*((3 + 2*Sqrt[2])/2)*Log[Pi] // Re; RealDigits[gamma1[1/8], 10, 104] // First
(* or, from version 7 up: *) RealDigits[StieltjesGamma[1, 1/8], 10, 104] // First

A255189 Decimal expansion of gamma_1(1/12), the first generalized Stieltjes constant at 1/12 (negated).

Original entry on oeis.org

2, 9, 8, 4, 2, 8, 7, 8, 2, 3, 2, 0, 4, 1, 3, 3, 1, 3, 0, 3, 3, 5, 1, 0, 2, 0, 2, 6, 0, 7, 5, 9, 2, 6, 3, 2, 3, 9, 8, 9, 2, 0, 4, 4, 0, 0, 1, 8, 6, 1, 0, 0, 5, 6, 8, 7, 0, 3, 6, 1, 0, 6, 7, 8, 3, 0, 9, 3, 3, 3, 8, 8, 5, 1, 5, 6, 1, 2, 3, 1, 6, 1, 4, 6, 4, 6, 2, 5, 1, 2, 7, 6, 9, 7, 0, 1, 2, 4, 2, 3, 4, 8, 7, 8

Offset: 2

Jean-François Alcover, Feb 16 2015

			-29.842878232041331303351020260759263239892044001861...

G. C. Greubel, Table of n, a(n) for n = 2..5002
Iaroslav V. Blagouchine, A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations, Journal of Number Theory (Elsevier), Volume 148, pages 537-592, March 2015 (arXiv preprint).
Eric Weisstein's MathWorld, Hurwitz Zeta Function.
Eric Weisstein's MathWorld, Stieltjes Constants.
Wikipedia, Stieltjes constants

Cf. A001620 (gamma), A082633 (gamma_1), A254327 (gamma_1(1/2)), A254331 (gamma_1(1/3)), A254345 (gamma_1(2/3)), A254347 (gamma_1(1/4)), A254348 (gamma_1(3/4)), A254349 (gamma_1(1/6)), A254350 (gamma_1(5/6)), A251866 (gamma_1(1/5)) A255188 (gamma_1(1/8)).

gamma1[1/12] = StieltjesGamma[1] + Sqrt[3]*(Derivative[2, 0][Zeta][0, 1/12] + Derivative[2, 0][Zeta][0, 11/12]) + 4*Pi*LogGamma[1/4] + 3*Pi*Sqrt[3]*LogGamma[1/3] - (((2 + Sqrt[3])/2)*Pi + (3/2)*Log[3] - Sqrt[3]*(1 - Sqrt[3])*Log[2] + 2*Sqrt[3]*Log[1 + Sqrt[3]])*EulerGamma - 2*Sqrt[3]*(3*Log[2] + Log[3] + Log[Pi])* Log[1 + Sqrt[3]] - ((7 - 6*Sqrt[3])/2)*Log[2]^2 - (3/4)*Log[3]^2 + 3*Sqrt[3]*((1 - Sqrt[3])/2)*Log[3]*Log[2] + Sqrt[3]*Log[2]*Log[Pi] - Pi*((17 + 8*Sqrt[3])/(2*Sqrt[3]))*Log[2] + ((Pi*(1 - Sqrt[3])*Sqrt[3])/4)*Log[3] - Pi*Sqrt[3]*(2 + Sqrt[3])*Log[Pi] // Re; RealDigits[gamma1[1/12], 10, 104] // First
(* or, from version 7 up: *) RealDigits[StieltjesGamma[1, 1/12], 10, 104] // First

cp's OEIS Frontend

A251866 Decimal expansion of gamma_1(1/5), the first generalized Stieltjes constant at 1/5 (negated).

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A254349 Decimal expansion of gamma_1(1/6), the first generalized Stieltjes constant at 1/6 (negated).

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A254331 Decimal expansion of gamma_1(1/3), the first generalized Stieltjes constant at 1/3 (negated).

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A254345 Decimal expansion of gamma_1(2/3), the first generalized Stieltjes constant at 2/3 (negated).

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A254347 Decimal expansion of gamma_1(1/4), the first generalized Stieltjes constant at 1/4 (negated).

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A254348 Decimal expansion of gamma_1(3/4), the first generalized Stieltjes constant at 3/4 (negated).

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A254350 Decimal expansion of gamma_1(5/6), the first generalized Stieltjes constant at 5/6 (negated).

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A255188 Decimal expansion of gamma_1(1/8), the first generalized Stieltjes constant at 1/8 (negated).

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