Generalized series involving harmonic numbers

Question

I was wondering whether is possible to find a closed form for the series $$\Xi(s)=\sum_{n\geq1} \frac{H_n}{n^s}$$ with $s\in(1,\infty)$ to let the series converge, or, at least with $s\in\mathbb{N}\setminus\{1\}$. Since $\Xi(2)=2\zeta(3)$ and $\Xi(3)=\frac{\zeta^2(2)}2$, I suppose $\Xi(s\in\mathbb{N}\setminus\{1\})$ can all be expressed in terms of Riemann $\zeta(z)$ function. The problem is that, while for $s=2$ we work with $\mathrm{Li}_2(x)$ and for $s=3$ it's all about reindexing, using these methods for $s\geq4$ becomes quite difficult. I'm sure I lack of some polylog's important identities, so I ask for your help: have you some idea to deduce a closed form of $\Xi(s)$?