How does $\lim\limits_{Re(s)\rightarrow -\infty} Li_s(e^{w})$ become $\Gamma(1-s)(-w)^{s-1}$?

Question

I am trying to prove the following property of polylogarithm. $$\lim\limits_{Re(s)\rightarrow -\infty} Li_s(e^{w}) = \Gamma(1-s)(-w)^{s-1} \text{ for } -\pi < \Im(w) < \pi.$$ As per Wikipedia's "Polylogarithm" entry, the above limit follows from the relationship of polylogarithm with Hurwitz-Zeta function as given below. $$\displaystyle \operatorname {Li}_{s}(z)={\Gamma (1-s) \over (2\pi )^{1-s}}\left[i^{1-s}~\zeta \left(1-s,~{\frac {1}{2}}+{\ln(-z) \over {2\pi i}}\right)+i^{s-1}~\zeta \left(1-s,~{\frac {1}{2}}-{\ln(-z) \over {2\pi i}}\right)\right]$$ How do I observe this? Also is the above limit uniform for all $w$ by any chance?