Are there examples of Mellin transforms of functions $ \int_{0}^{\infty}f(x)x^{s-1}\mathrm dx$ that have only real zeroes or have only zeroes on the critical line?
For example, the Mellin transform of the Hermite polynomials has zeroes on the critical line $\mathrm{Re}(s)=1/2$. Can this fact be extended to the Mellin transform of any function $ f(x)$ which is its own Fourier transform ? $ F(f(x))=kf(x) $