The table of integrals [35] contains many entries that are expressible in terms of the polylogarithm function

(9.1.1) Lis(z) := ∞∑ k=1


ks .

In this paper we describe the evaluation of some of them. The series (9.1.1) converges for |z| < 1 and Re s > 1. The integral representation

(9.1.2) Lis(z) = z


xs−1 dx ex − z

provides an analytic extension to C. Here Γ(s) is the classical gamma function defined by

(9.1.3) Γ(s) :=

xs−1e−x dx.