RIemann-Weyl formula and application

mhill
given a real function 'f' so the Fourier transform of f(x) exists and assuming Riemann Hypothesis is true so all the non-trivial zeros of Riemann Zeta are of the form 1/2+it then the Riemann-Weyl formula

$$\sum _{t} f(t) = f(i/2)+f(-i/2)+2g(0)\pi- (2\pi)^{-1} \int_{-\infty}^{\infty}dr f(r) \frac{ \Gamma '(1/4+ir/2) }{\Gamma (1/4+ir/2)}+ \sum _{n=1}^{\infty} \Lambda (n) (n)^ {-1/2}g(lon)$$

the first sum run over the non-trivial zeros, my question is if all the zeros are 1/2+it with 't' real number , the only condition for Riemann-Weyl formula to converge is that the Fourier transform of f(x) , which can be a function or a distribution must exists ? , under what conditions can be apply Riemann-Weyl summation formula? , if f(x) is a function g(u) is its Fourier transform

for example if i wan to obtain the sum of $$e^{-t^{2}}$$ where 't' runs over the imaginary part of all Zeta zeros in the form $$\zeta (1/2+it)=0$$ , assuming RH is real could we use the Riemann-Weyl formula ? that was all thanks.