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Decimal expansion of zeta'(-1, 1/6).
2

%I #13 Mar 26 2019 11:46:02

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

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

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

%N Decimal expansion of zeta'(-1, 1/6).

%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 25.

%F Equals -Pi/(12*sqrt(3)) + log(2)/72 + log(3)/144 + PolyGamma(1, 1/3)/(8*sqrt(3)*Pi) + Zeta'(-1)/6.

%F A324997 + A324998 = log(2)/36 + log(3)/72 + Zeta'(-1)/3.

%e 0.070452592367204142475462166806035927785155027545830206477019332868362...

%p evalf(Zeta(1,-1,1/6), 120);

%p evalf(-Pi/(12*sqrt(3)) + log(2)/72 + log(3)/144 + Psi(1, 1/3)/(8*sqrt(3)*Pi) + Zeta(1,-1)/6, 120);

%t RealDigits[Derivative[1, 0][Zeta][-1, 1/6], 10, 120][[1]]

%t N[With[{k=1}, -(9^k - 1) * (2^(2*k-1) + 1) * BernoulliB[2*k] * Pi/(8*Sqrt[3]*k*6^(2*k - 1)) + BernoulliB[2*k] * (3^(2*k-1) - 1)*Log[2]/(4*k*6^(2*k - 1)) + BernoulliB[2*k]*(2^(2*k-1) - 1) * Log[3]/(4*k*6^(2*k-1)) - (-1)^k*(2^(2*k-1) + 1) * PolyGamma[2*k-1,1/3] / (2*Sqrt[3]*(12*Pi)^(2*k - 1))+(2^(2*k - 1) - 1)*(3^(2*k - 1) - 1)*Zeta'[1-2*k]/2/6^(2*k-1)], 120]

%o (PARI) zetahurwitz'(-1, 1/6) \\ _Michel Marcus_, Mar 24 2019

%Y Cf. A084448, A240966, A324998.

%K nonn,cons

%O 0,2

%A _Vaclav Kotesovec_, Mar 23 2019