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

%I A254350 #21 Feb 16 2025 08:33:24
%S A254350 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,
%T A254350 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,
%U A254350 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
%N A254350 Decimal expansion of gamma_1(5/6), the first generalized Stieltjes constant at 5/6 (negated).
%H A254350 G. C. Greubel, <a href="/A254350/b254350.txt">Table of n, a(n) for n = 0..5000</a>
%H A254350 Iaroslav V. Blagouchine, <a href="http://arxiv.org/abs/1401.3724">A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments</a>, arXiv:1401.3724 [math.NT], 2015.
%H A254350 Iaroslav V. Blagouchine, <a href="http://www.researchgate.net/publication/259743013_A_theorem_for_the_closed-form_evaluation_of_the_first_generalized_Stieltjes_constant_at_rational_arguments_and_some_related_summations">A theorem ... (same title)</a>, Journal of Number Theory Volume 148, March 2015, Pages 537-592.
%H A254350 Iaroslav V. Blagouchine, <a href="https://doi.org/10.1007/s11139-013-9528-5">Rediscovery of Malmsten’s integrals, their evaluation by contour integration methods and some related results</a>, The Ramanujan Journal October 2014, Volume 35, Issue 1, pp 21-110.
%H A254350 Iaroslav V. Blagouchine, <a href="http://www.researchgate.net/publication/257381156_Rediscovery_of_Malmsten%27s_integrals_their_evaluation_by_contour_integration_methods_and_some_related_results">Rediscovery of Malmsten’s integrals: Full PDF text</a>.
%H A254350 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HurwitzZetaFunction.html">Hurwitz Zeta Function</a>.
%H A254350 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/StieltjesConstants.html">Stieltjes Constants</a>.
%H A254350 Wikipedia, <a href="http://en.wikipedia.org/wiki/Stieltjes_constants">Stieltjes constants</a>
%F A254350 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).
%e A254350 -0.24616900381139073314849171532749069577086909012844...
%t A254350 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] -
%t A254350       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
%t A254350 (* Or, from Mma version 7 up: *) RealDigits[StieltjesGamma[1, 5/6], 10, 104] // First
%Y A254350 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)).
%K A254350 nonn,cons,easy
%O A254350 0,1
%A A254350 _Jean-François Alcover_, Jan 29 2015