I have posted this question on MO, and I didn't get an answer. We have the following integral:
$$I(s)=\int_{0}^{\infty} \frac{s}{2x}\left(E_{s/2}((\pi x)^{s/2})-1\right)\omega(x)dx -\lim_{z \to 1 }\zeta(z)$$
where $E_{\alpha}(z)$ is the Mittag-Leffler function,
$\omega(x)=\dfrac{\theta(ix)-1}{2}$,
$\theta(x)$ is the Jacobi theta function,
and $\zeta(s)$ is the Riemann zeta function.
The integral behaves well for $Re(s)>1$. I am trying to extend the domain of the integral to the whole complex plane except for some points. I have tried the identities:
$\omega(x^{-1})=-\frac{1}{2}+\frac{1}{2}x^{1/2}+x^{1/2}\omega(x)$,
and
$E_{\alpha}(z^{-1})=1-E_{-\alpha}(z)$,
to split the integration at $1$, and replace $x$ by $x^{-1}$ in the interval $[0,1]$, and I had hoped to get a factor that would cancel out with the divergent $\zeta(z)$, and get the Euler–Mascheroni constant or something. But it seems I am lost!!! Hence, the post.