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A324994 Decimal expansion of zeta'(-1, 2/3) (negated).

%I A324994 #17 Feb 16 2025 08:33:58
%S A324994 0,1,3,9,6,2,4,4,7,3,1,2,3,7,0,7,4,3,8,8,0,3,4,4,6,0,4,4,4,1,4,0,9,2,
%T A324994 6,3,9,8,8,5,7,6,6,5,9,9,8,8,1,2,4,3,1,7,1,8,4,8,4,1,3,9,7,5,7,4,9,0,
%U A324994 3,3,7,2,9,8,4,8,3,3,2,6,2,8,5,6,2,5,6,4,5,3,5,5,4,2,4,9,7,0,3,6,2,1,5,1,0,6
%N A324994 Decimal expansion of zeta'(-1, 2/3) (negated).
%H A324994 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 A324994 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HurwitzZetaFunction.html">Hurwitz Zeta Function</a>, formula 23
%F A324994 Equals Pi/(18*sqrt(3)) - log(3)/72 - PolyGamma(1, 1/3) / (12*sqrt(3)*Pi) - Zeta'(-1)/3.
%F A324994 A324993 + A324994 = -log(3)/36 - 2*Zeta'(-1)/3.
%e A324994 -0.01396244731237074388034460444140926398857665998812431718484139757490...
%p A324994 evalf(Zeta(1,-1,2/3), 120);
%p A324994 evalf(Pi/(18*sqrt(3)) - log(3)/72 - Psi(1, 1/3) / (12*sqrt(3)*Pi) - Zeta(1,-1)/3, 120);
%t A324994 RealDigits[Derivative[1, 0][Zeta][-1, 2/3], 10, 120][[1]]
%t A324994 N[With[{k=1}, Sqrt[3] * (9^k - 1) * BernoulliB[2*k] * Pi / (9^k * 8*k) - 3*BernoulliB[2*k] * Log[3] / 9^k / 4 / k + (-1)^k * PolyGamma[2*k-1,1/3] / 2 / Sqrt[3] / (6*Pi)^(2*k-1) - (9^k-3)*Zeta'[-2*k+1]/2/9^k], 120]
%o A324994 (PARI) zetahurwitz'(-1, 2/3) \\ _Michel Marcus_, Mar 24 2019
%Y A324994 Cf. A084448, A240966, A324993.
%K A324994 nonn,cons
%O A324994 0,3
%A A324994 _Vaclav Kotesovec_, Mar 23 2019