This question related to this question in SE ,I would like to know how do I

evaluate this sum for $s$ is a complex variable :$$\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{2s}n!}$$ .

**Edit01**:And I think the General complex solution of $$\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{2s}n!}=0$$ would be $s=-1+i\beta$ for some values of $\beta$ and for $\beta =0$ it's a trivial zero

**Note 01** :In wolfram alpha the series is converge but i don't know if it has a nice closed form and really the convergence in complex number ensemble is not clear !!!

**Note 02** I edited the question only for the zeros of this series since it's convergent after some computation in wolfram alpha!!!

Thank you for any help