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A324992 Decimal expansion of zeta'(-1, 1/2).

%I A324992 #25 Feb 16 2025 08:33:58
%S A324992 0,5,3,8,2,9,4,3,9,3,2,6,8,9,4,4,1,0,0,4,7,9,0,8,4,9,1,7,2,7,2,9,9,6,
%T A324992 3,1,0,4,5,5,3,9,0,1,7,9,0,2,5,9,0,2,5,6,2,4,4,8,9,9,4,8,6,1,1,6,4,5,
%U A324992 5,1,1,5,5,8,4,5,5,1,3,0,6,5,6,2,8,5,1,5,7,8,2,0,8,0,7,0,2,6,5,7,8,8,2,7,1,8
%N A324992 Decimal expansion of zeta'(-1, 1/2).
%H A324992 J. Miller and V. Adamchik, <a href="https://doi.org/10.1016/S0377-0427(98)00193-9">Derivatives of the Hurwitz Zeta Function for Rational Arguments</a>, Journal of Computational and Applied Mathematics 100 (1998) 201-206. [contains a large number of typos]
%H A324992 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HurwitzZetaFunction.html">Hurwitz Zeta Function</a>, formula 22.
%F A324992 Equals -log(2)/24 - Zeta'(-1)/2 = A261829 - log(2)/24.
%F A324992 Equals -1/24 - log(2)/24 + log(A)/2, where A is the Glaisher-Kinkelin constant A074962.
%F A324992 Equals (log(Pi) - 1 + gamma)/24 - Zeta'(2)/(4*Pi^2), where gamma is the Euler-Mascheroni constant A001620.
%e A324992 0.053829439326894410047908491727299631045539017902590256244899486116455...
%p A324992 evalf(Zeta(1,-1,1/2), 120);
%p A324992 evalf(-log(2)/24 - Zeta(1,-1)/2, 120);
%t A324992 RealDigits[Derivative[1, 0][Zeta][-1, 1/2], 10, 120][[1]]
%t A324992 N[With[{k=1}, -BernoulliB[2*k] * Log[2] / 4^k / k - (2^(2*k - 1) - 1) * Zeta'[1 - 2*k] / 2^(2*k - 1)], 120]
%o A324992 (PARI) zetahurwitz'(-1, 1/2) \\ _Michel Marcus_, Mar 24 2019
%Y A324992 Cf. A074962, A084448, A240966, A261829, A306635.
%K A324992 nonn,cons
%O A324992 0,2
%A A324992 _Vaclav Kotesovec_, Mar 23 2019