calculus - Identification of a function: $sum_{k=1}^infty(log(k))^nfrac{z^k}{k}$

I recently came across the following function

$$\sum_{k=1}^\infty(\log(k))^n\frac{z^k}{k}$$

I found it while dealing with the polylogarithm function, $Li_n (z)$ (Notice that if instead of $(\log(k))^n$ we had $k^n$ then the above expression would become $Li_{1-n}(z)$. Still these functions are quite different.)

I was wondering if this function is known, and if there are good numerical approximations to estimate it?

Thank you in advance for your help.