Inverse Mellin transform and Bessel function

Question

I am calculating the inverse Mellin transform of the function $$\frac{e^{-s^2}}{s}, \ s>0.$$ I tried using Mathematica and it gives me $$\mathcal{M}^{-1}\left\{\frac{e^{-s^2}}{s} \right\}(x) = J_0(2\sqrt{-\log x})u(1-x), $$ where $J_0$ is the Bessel function of the first kind and $u$ is the heaviside function. However, I tried to compute the Mellin transform of the above and it gave me $e^{-\frac{1}{s}}/s $ instead. By a direct computation, I found that the above expression is indeed the inverse Mellin transform of $e^{-\frac{1}{s}}/s $. My question is then to look for the actual inverse Mellin transform of $\frac{e^{-s^2}}{s}$. At least, can we show that the inverse Mellin transform of the above exists and if it is real-valued? Any discussions or hints are appreciated.