It's known that when you analytically continue the Riemann zeta function to the complex plane, it has complex roots on a vertical line.
I would like to figure out if there are complex roots to the following analytic continuation of the function $\varphi(s)=\sum_{n\ge1} e^{-n^s}:$
$$ \varphi(s)=\Gamma\left(1+\frac{1}{s}\right)+\sum_{n\ge0} \frac{(-1)^n}{n!}\zeta(-ns) ~~~~~~~~~~~~(*)$$
Are there any complex roots for $(*)?$
What does a complex plot of this function look like in a circular region centered at the origin with radius $r=30?$
$(*)$ converges for complex $s\ne0$ and for $\Re(s)<1$.
Since $(*)$ is quite complicated I'm having trouble analyzing it.