The functional equation for the Hurwitz Zeta function ζ(s, a) is used to obtain formulas for derivatives of ζ(s, a) at negative odd s and rational a. For several of these rational arguments, closed-form expressions are given in terms of simpler transcendental functions, like the logarithm, the polygamma function, and the Riemann Zeta function. © 1998 Elsevier Science B.V. All rights reserved.
Miller, J., & Adamchik, V. S. (1998). Derivatives of the Hurwitz Zeta function for rational arguments. Journal of Computational and Applied Mathematics, 100(2), 201–206. https://doi.org/10.1016/S0377-0427(98)00193-9