On two sided decaying exponential functions and Xi function

It is well known that a real two-sided decaying exponential function $g_0(t) = e^{b t} u(-t) +  e^{-a t} u(t) $, does not have zeros in its Fourier Transform, where $u(t)$ is Heaviside unit step function and $a, b > 0$ are real. We consider Xi function $\xi(s)$ which is evaluated at $s = \frac{1}{2} + \sigma + i \omega$, given by $\xi(\frac{1}{2} + \sigma + i \omega)= E_{p\omega}(\omega)$, where $\sigma, \omega$ are real and $-\infty \leq \omega \leq \infty$ and compute its inverse Fourier transform given by $E_p(t)$, which is expressed as an \textbf{infinite summation of two-sided decaying exponential functions} using Taylor series expansion.