Let $s=\sigma+it$ be a complex number and define the function:
$$F(s)=\sum_{k=2}^{\infty}\frac{p_\pi(k)}{k^s}$$
Where $p_\pi(k)$ is the number of unordered factorizations of $k$, corresponding to OEIS A001055
This is a Dirichlet series that converges for $\sigma>1$.
I know that finding the analytic continuation for any Dirichlet series is absolutely not trivial, and many of them require more knowledge about the nontrivial zeros of $\zeta(s)$. The function above is just an example of a function I am working with.
Is it possible to find the analytic continuation of the function above, with the known methods? If not, why?
Do we know any other analytic continuation for the Dirichlet series out of Dirichlet's L-Functions?