Has anyone studied a function of the form $$\eta(s) = \sum_{n=1}^{\infty} \frac{1}{\Gamma(n)^{s}} = \sum_{n=0}^{\infty}\frac{1}{k!^s}$$ This series is appearing in my research on the volumetric properties of infinite families of polytopes and I am not sure how to evaluate the sum, or if this is a well-known function other than the fact that $\eta(2) = I_{0}(2)$, where $I_{\nu}(x)$ is the modified Bessel function of the first kind as seen here. I am interested in evaluating this series for any $s \in (1,\infty)$. For integer values of $s$ is it equal to a generalized hypergeometric function?

EDIT: It would be sufficient for my purposes to prove that $\eta(s)$ is finite for any $s \in (0,\infty)$, any ideas?