The Riemann $\Xi(z)$ function admits a Fourier transform of a even kernel $\Phi(t)=3e^{5t/4}\theta'(e^{t})+2e^{9t/4}\theta''(e^{t})$. Here $\theta(z)$ is the Jacobi theta function.
$$\theta(z)=\sum_{n=-\infty}^{\infty}\exp(-\pi n^2 z),\qquad Re(z)>0,\tag{1}$$
$$\Phi(t)=\sum_{n=1}^{\infty}\left(2\pi^2n^4\exp(9t/4)-3\pi n^2\exp(5t/4)\right)\exp\left(-\pi n^2 \exp(t)\right)\tag{2}$$
Let $f_m(t)$ be the partial summation of $$f_m(t)=\sum_{n=1}^{m}\left(2\pi^2n^4\exp(9t/4)-3\pi n^2\exp(5t/4)\right)\exp\left(-\pi n^2 \exp(t)\right)\tag{3}$$
Question (1): How to prove that $\Phi(t)$ is divergent when $Im(t)=\pi/2$?
Question (2): Is $f_m(t)$ convergent when $t \in \mathbb{C}$?
Any reference and comment are welcome. -mike