Regularity of root spacing of $G(z)=\sum_{n=1}^{\infty} \frac{e^{-n^{2}}}{n^{z}}$ Define, on $\mathbb{C}$:
$$G(z)=\sum_{n=1}^{\infty} \frac{e^{-n^{2}}}{n^{z}}$$
A domain colored portrait of $G(z)$ (boxes are supposed to be negative signs):

suggests that the roots of $G(z)$ are equally spaced along lines of fixed real component. So:


*

*Are the roots of $G(z)$ regularly spaced for some fixed $\Re(z)$? and if so:

*Are there analytic expressions for the roots?

*Can $G(z)$ expressed as a Weierstrass product?


Approximate $G(z)$ by the first two terms, $g(z) := e^{-1} + e^{-4}/2^z$. The zeroes of $g$ are
$$- \frac{3}{\log 2} + (2k+1) \frac{\pi}{\log 2} i \approx - 4.328 + (2k+1) 4.532 i \ \mathrm{for} \ k \in \mathbb{Z}.$$
In particular, the zeroes of $g$ are perfectly regularly spaced.
Your zeroes are pretty close to these points, so I suspect that the regularity you are seeing just means that the zeroes of $G$ are pretty close to those of $g$.
