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

%I #20 Mar 26 2019 10:56:46

%S 0,9,3,7,2,6,2,0,1,7,6,0,7,7,9,4,2,7,4,8,4,2,0,0,8,9,9,1,3,3,1,9,2,8,

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

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

%N Decimal expansion of zeta'(-1, 1/3).

%H 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HurwitzZetaFunction.html">Hurwitz Zeta Function</a>, formula 23

%F Equals -Pi/(18*sqrt(3)) - log(3)/72 + PolyGamma(1, 1/3) / (12*sqrt(3)*Pi) - Zeta'(-1)/3.

%F A324993 + A324994 = -log(3)/36 - 2*Zeta'(-1)/3.

%e 0.093726201760779427484200899133192867368837286938738021525448092545434...

%p evalf(Zeta(1,-1,1/3), 120);

%p evalf(-Pi/(18*sqrt(3)) - log(3)/72 + Psi(1, 1/3) / (12*sqrt(3)*Pi) - Zeta(1, -1)/3, 120);

%t RealDigits[Derivative[1, 0][Zeta][-1, 1/3], 10, 120][[1]]

%t 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 (PARI) zetahurwitz'(-1, 1/3) \\ _Michel Marcus_, Mar 24 2019

%Y Cf. A084448, A240966, A306651, A324994.

%K nonn,cons

%O 0,2

%A _Vaclav Kotesovec_, Mar 23 2019