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%I #15 Dec 01 2020 11:35:15
%S 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,
%T 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,
%U 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
%N Decimal expansion of gamma_1(1/8), the first generalized Stieltjes constant at 1/8 (negated).
%H G. C. Greubel, <a href="/A255188/b255188.txt">Table of n, a(n) for n = 2..5002</a>
%H 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 Iaroslav V. Blagouchine, <a href="http://dx.doi.org/10.1016/j.jnt.2014.08.009">A theorem ... (same title)</a>, Journal of Number Theory Volume 148, March 2015, Pages 537-592.
%H Iaroslav V. Blagouchine, <a href="http://dx.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 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HurwitzZetaFunction.html">Hurwitz Zeta Function</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/StieltjesConstants.html">Stieltjes Constants</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Stieltjes_constants">Stieltjes constants</a>
%e -16.641719763609315662841926230373944928513266065474455...
%t 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
%t (* or, from version 7 up: *) RealDigits[StieltjesGamma[1, 1/8], 10, 104] // First
%Y 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)).
%K nonn,cons,easy
%O 2,2
%A _Jean-François Alcover_, Feb 16 2015
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