  We examine the use of the Euler-Maclaurin formula and new derived uniform
asymptotic expansions for the numerical evaluation of the Lerch transcendent
$\Phi(z, s, a)$ for $z, s, a \in \mathbb{C}$ to arbitrary precision. A detailed
analysis of these expansions is accompanied by rigorous error bounds. A
complete scheme of computation for large and small values of the parameters and
argument is described along with algorithmic details to achieve high
performance. The described algorithm has been extensively tested in different
regimes of the parameters and compared with current state-of-the-art codes. An
open-source implementation of $\Phi(z, s, a)$ based on the algorithms described
in this paper is available.
