  We study the use of the Euler-Maclaurin formula to numerically evaluate the
Hurwitz zeta function $\zeta(s,a)$ for $s, a \in \mathbb{C}$, along with an
arbitrary number of derivatives with respect to $s$, to arbitrary precision
with rigorous error bounds. Techniques that lead to a fast implementation are
discussed. We present new record computations of Stieltjes constants, Keiper-Li
coefficients and the first nontrivial zero of the Riemann zeta function,
obtained using an open source implementation of the algorithms described in
this paper.
