%I #16 Mar 26 2019 10:57:19

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

%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 -0.01396244731237074388034460444140926398857665998812431718484139757490...

%p evalf(Zeta(1,-1,2/3), 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, 2/3], 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, 2/3) \\ _Michel Marcus_, Mar 24 2019

%Y Cf. A084448, A240966, A324993.

%K nonn,cons

%O 0,3

%A _Vaclav Kotesovec_, Mar 23 2019