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The dilogarithm is the order-two polylogarithm, defined for $|z|<1$ by ${\mathrm{Li}}_2(z)={\sum }_{n\ge 1}{z}^n/n^2$. Its derivative is $-\mathrm{Log}(1-z)/z$, with a removable value at zero; a chosen complex logarithm branch gives its analytic continuation.

For fixed $s$, the factor $n^{-s}$ grows at most polynomially in $n$, whereas $|z{|}^n$ decays exponentially when $|z|<1$. The defining series for the polylogarithm therefore converges absolutely and locally uniformly for every complex $s$ and $|z|<1$. It is holomorphic in $z$ on the unit disc and entire in $s$ there.