Hi, I've just posted a paper describing an efficient algorithm for computing the (classical, fractional) polylogarithm Li_s(z) for arbitrary complex values of s and z; and similarly for the the Hurwitz zeta function.
http://arxiv.org/abs/math.CA/0702243 The algorithm generalizes Borwein's "An Efficient Algorithm for the Riemann zeta function", and, draws on Cohen, Villegas and Zagier "Convergence Acceleration for Alternating Series" to propose a way of accelerating general oscillatory series. The paper also provides a low-brow review of the monodromy of the polylogarithm, as there does not appear to be any simple discussion in the literature. The algo is implemented in GMP, and the source code is available upon request, under the LGPL license. --linas
