A075700 -id:A075700 - cp's OEIS frontend

A102887 Decimal expansion of Integral_{x=0..1} log(gamma(x))^2 dx.

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

Offset: 1

Eric W. Weisstein, Jan 15 2005

Also equals (1/6)*log(2*Pi)^2 + 2*log(A)*log(2*Pi) - (1/6)*gamma*log(2*Pi) + Pi^2/48 + 2*gamma*log(A) + zeta''(2)/(2*Pi^2) (with A the Glaisher-Kinkelin constant). - Jean-François Alcover, Apr 29 2013

			1.8663170837935620809929679379782897398...

George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press (2006), p. 236.

G. C. Greubel, Table of n, a(n) for n = 1..10000
M. L. Glasser, A note on Beukers's and related integrals, Amer. Math. Monthly 126(4) (2019), 361-363.

Cf. A001620, A074962, A075700, A201994.

EulerGamma^2/12 + Pi^2/48 + (EulerGamma*Log[2*Pi])/6 + Log[2*Pi]^2/3 - ((EulerGamma + Log[2*Pi])*Zeta'[2])/Pi^2 + Zeta''[2]/(2*Pi^2)

intnum(x=0,1, log(gamma(x))^2) \\ Michel Marcus, Aug 27 2015

Equals gamma^2/12 + Pi^2/48 + (gamma*log(2*Pi))/6 + log(2*Pi)^2/3 - ((gamma + log(2*Pi))*zeta'(2))/Pi^2 + zeta''(2)/(2*Pi^2).

Equals -Integral_{x=0..1, y=0..1} log(gamma(x*y))^2/log(x*y) dx dy. (Apply Theorem 1 or Theorem 2 in Glasser (2019).) - Petros Hadjicostas, Jun 30 2020

A257549 Decimal expansion of zeta''(0) (negated).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Apr 29 2015

Essentially the same as A245273. - R. J. Mathar, Apr 30 2015

			2.00635645590858485121010002672996043819899491016091988116986828...

Tom M. Apostol, Formulas for higher derivatives of the Riemann zeta function, Mathematics of Computation 44 (1985), p. 223-232.
Richard E. Crandall, Unified algorithms for polylogarithm, L-series, and zeta variants. p. 15.
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 104.
Eric Weisstein's World of Mathematics, Riemann Zeta Function.
Eric Weisstein's World of Mathematics, Stieltjes Constants.

Cf. A075700, A082633, A261508, A201994.

evalf(-Zeta(2, 0), 120); # Vaclav Kotesovec, Apr 29 2015

RealDigits[ StieltjesGamma[1] + EulerGamma^2/2 - Pi^2/24 - (1/2)*(Log[2] + Log[Pi])^2, 10, 104] // First

-zeta''(0) \\ Charles R Greathouse IV, Mar 10 2016

zeta''(0) = gamma_1 + gamma^2/2 - Pi^2/24 - (1/2)*(log(2)+log(Pi))^2, where gamma_1 is the first Stieltjes constant.

A258162 Decimal expansion of the log Gamma integral LG_3 = Integral_{0..1} log(Gamma(x))^3 dx.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, May 22 2015

			5.7403888072294742800195716881024614629610130074548733314254...

G. C. Greubel, Table of n, a(n) for n = 1..1000
David H. Bailey, David Borwein, and Jonathan M. Borwein, On Eulerian Log-Gamma Integrals and Tornheim-Witten Zeta Functions.

Cf. A075700 (LG_1), A102887 (LG_2), A258163 (LG_4), A258164 (LG_5).

evalf(Int(log(GAMMA(x))^3,x=0..1),120); # Vaclav Kotesovec, May 22 2015

LG3 = NIntegrate[LogGamma[x]^3, {x, 0, 1}, WorkingPrecision -> 104]; RealDigits[LG3] // First

intnum(x=0, 1, log(gamma(x))^3) \\ Michel Marcus, Oct 24 2017

A122914 Decimal expansion of (1 + log(2*Pi))/2, the entropy of the standard normal distribution.

Original entry on oeis.org

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

Offset: 1

Joseph Biberstine (jrbibers(AT)indiana.edu), Sep 18 2006

For a normal distribution with standard deviation sigma, add log(sigma). - Stanislav Sykora, Jan 15 2017

			1.4189385332046727417803297364056176398613974736377834128171515404827656959...

Wikipedia, Normal distribution.

Partial quotients in A122915.

Cf. A061444, A075700.

Mathematica
```
RealDigits[(1 + Log[2 Pi])/2, 10, 80]
```

Equals (1 + log(2*Pi))/2 = 1/2 - A075700 = (1 + A061444)/2.
Equals -zeta(0) - zeta'(0). - Peter Luschny, May 16 2020
Equals 1 + G'(1), where G(x) is the Barnes G-function. - Amiram Eldar, Jun 08 2022

a(80) corrected by Georg Fischer, Jul 10 2021

A258163 Decimal expansion of the log Gamma integral LG_4 = Integral_{0..1} log(Gamma(x))^4 dx.

Original entry on oeis.org

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

Offset: 2

Jean-François Alcover, May 22 2015

			23.389514465516801619600559105059140659007527683198464667785452...

David H. Bailey, David Borwein, and Jonathan M. Borwein, On Eulerian Log-Gamma Integrals and Tornheim-Witten Zeta Functions.

Cf. A075700 (LG_1), A102887 (LG_2), A258162 (LG_3), A258164 (LG_5).

evalf(Int(log(GAMMA(x))^4,x=0..1),120); # Vaclav Kotesovec, May 22 2015

LG4 = NIntegrate[LogGamma[x]^4, {x, 0, 1}, WorkingPrecision -> 105];
RealDigits[LG4] // First

A258164 Decimal expansion of the log Gamma integral LG_5 = Integral_{0..1} log(Gamma(x))^5 dx.

Original entry on oeis.org

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

Offset: 3

Jean-François Alcover, May 22 2015

			118.2987931845512587541672907292964484902928529010820657473411046...

David H. Bailey, David Borwein, and Jonathan M. Borwein, On Eulerian Log-Gamma Integrals and Tornheim-Witten Zeta Functions.

Cf. A075700 (LG_1), A102887 (LG_2), A258162 (LG_3), A258163 (LG_4).

evalf(Int(log(GAMMA(x))^5,x=0..1),120); # Vaclav Kotesovec, May 22 2015

LG5 = NIntegrate[LogGamma[x]^5, {x, 0, 1}, WorkingPrecision -> 105]; RealDigits[LG5] // First

A261508 Decimal expansion of -zeta'''(0).

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Aug 22 2015

			6.004711166862254447761060813366375285461807668295980132893...

Tom M. Apostol, Formulas for higher derivatives of the Riemann zeta function, Mathematics of Computation 44 (1985), pp. 223-232.
Eric Weisstein's World of Mathematics, Riemann Zeta Function
Eric Weisstein's World of Mathematics, Stieltjes Constants

Cf. A075700, A257549, A201995, A082633, A086279.

Maple
```
evalf(-Zeta(3, 0), 120);
```

RealDigits[3*Log[2*Pi]*StieltjesGamma[1] + 3*EulerGamma*StieltjesGamma[1] + 3/2*StieltjesGamma[2] - Zeta[3] - 1/2*Log[2*Pi]^3 - 1/8*Pi^2*Log[2*Pi] + 3/2*EulerGamma^2*Log[2*Pi] + EulerGamma^3, 10, 120][[1]]

-zeta'''(0) \\ Charles R Greathouse IV, Mar 10 2016

A122915 Partial quotients of (1 + Log[2 Pi])/2, the entropy of the standard normal distribution.

Original entry on oeis.org

1, 2, 2, 1, 1, 2, 2, 8, 1, 8, 4, 1, 4, 1, 3, 18, 1, 7, 1, 4, 2, 4, 2, 1, 2, 1, 4, 1, 1, 17, 1, 1, 1, 23, 1, 1, 2, 28, 3, 2, 4, 1, 2, 3, 1, 39, 12, 2, 1, 1, 120, 1, 6, 1, 5, 1, 1, 1, 1, 1, 1, 4, 3, 1, 1, 5, 1, 5, 14, 1, 3, 3, 1, 5, 1, 12, 2, 7, 2, 1

Offset: 0

Joseph Biberstine (jrbibers(AT)indiana.edu), Sep 18 2006

a = 1/2 - A075700; a = (1 + A061444)/2.

Decimal expansion in A122914.

ContinuedFraction[(1 + Log[2 Pi])/2, 80]

a = (1 + Log[2 Pi])/2

A248859 Decimal expansion of log(sqrt(2*Pi))/e, a constant appearing in the asymptotic expansion of (n!)^(1/n).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Mar 03 2015

			0.3380585940662399023702794509615188742685137583402...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020-2021, p. 57.
Shafiqur Rahman and Leonard Giugiuc, Problem 4285, Crux Mathematicorum, Vol. 43, No. 9 (2017), pp. 399 and 401; Solution to Problem 4285, ibid., Vol. 44, No. 9 (2018), p. 395.

Cf. A001113, A019762, A061444, A075700 (log(sqrt(2*Pi))).

SetDefaultRealField(RealField(100)); R:= RealField(); Log(2*Pi(R))/(2*Exp(1)); // G. C. Greubel, Oct 07 2018

RealDigits[Log[Sqrt[2*Pi]]/E, 10, 105] // First

log(2*Pi)/2/exp(1) \\ Charles R Greathouse IV, Apr 20 2016

Equals lim_{n -> infinity} (n!)^(1/n) - n/e - log(n)/(2*e).

Equals A075700/A001113 = A061444/A019762. - Amiram Eldar, Apr 12 2022

A271854 Decimal expansion of -zeta'(-1/2), negated derivative of the Riemann zeta function at -1/2.

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, Apr 23 2016

			zeta'(-1/2) = -0.36085433959994760734742080636395106588485278791863221...

Stanislav Sykora, Table of n, a(n) for n = 0..1000

Values of |zeta'(x)| for various x: A073002 (+2), A075700 (0), A084448 (-1), A114875 (+1/2), A240966 (-2), A244115(+3), A259068 (-3), A259069 (-4), A259070 (-5), A259071 (-6), A259072 (-7), A259073 (-8), A261506 (+4), A266260 (-9), A266261 (-10), A266262 (zeta'(-11)), A266263 (zeta'(-12)), A260660 (zeta'(-13)), A266264 (zeta'(-14)), A266270 (zeta'(-15)), A266271 (zeta'(-16)), A266272 (zeta'(-17)), A266273 (zeta'(-18)), A266274 (zeta'(-19)), A266275 (zeta'(-20)), A271521 (i).

RealDigits[N[-Zeta'[-1/2], 106]] [[1]] (* Robert Price, Apr 28 2016 *)

PARI
```
-zeta'(-1/2)
```

cp's OEIS Frontend

A102887 Decimal expansion of Integral_{x=0..1} log(gamma(x))^2 dx.

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A257549 Decimal expansion of zeta''(0) (negated).

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A258162 Decimal expansion of the log Gamma integral LG_3 = Integral_{0..1} log(Gamma(x))^3 dx.

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A122914 Decimal expansion of (1 + log(2*Pi))/2, the entropy of the standard normal distribution.

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A258163 Decimal expansion of the log Gamma integral LG_4 = Integral_{0..1} log(Gamma(x))^4 dx.

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A258164 Decimal expansion of the log Gamma integral LG_5 = Integral_{0..1} log(Gamma(x))^5 dx.

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A261508 Decimal expansion of -zeta'''(0).

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