This question related to [this](https://math.stackexchange.com/q/1575992/230303) 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 



How do I evaluate this sum for $s$ is a complex variable :$\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{2s}n!}$?
*Mathematica* says that 
$$\sum\_{n=1}^\infty \frac{(-1)^{n+1}}{n^k n!} = 
\, \_kF\_k(1,\dotsc, 1;2,\dotsc,2;-1),$$ where the numbers of $1$s (and $2$s) are both equal to $k.$ This suggests to me that there is no closed form.


