You are asking for the inverse Laplace transform $g_\beta(t)$ of the stretched exponential function, $f_\beta(s)=\exp(-(s/s_0)^\beta$. For $\beta=1$ this is a Dirac delta function, $g_1(t)=\delta(t-1/s_0)$, and for $\beta=1/2$ it is the Lévy distribution, $$g_{1/2}(t)=\exp\left(-\frac{1}{4 s_0 t}\right)\frac{1}{2t \sqrt{\pi s_0 t}}.$$ There are no expressions in terms of elementary functions for other values of $\beta$. A convenient integral representation, suitable for numerical evaluation, is given here, equation 29.

You are asking for the inverse Laplace transform $g_\beta(t)$ of the stretched exponential function, $f_\beta(s)=\exp[-(s/s_0)^\beta]$. For $\beta=1$ this is a Dirac delta function, $g_1(t)=\delta(t-1/s_0)$, and for $\beta=1/2$ it is the Lévy distribution, $$g_{1/2}(t)=\exp\left(-\frac{1}{4 s_0 t}\right)\frac{1}{2t \sqrt{\pi s_0 t}}.$$ There are no expressions in terms of elementary functions for other values of $\beta$. A convenient integral representation, suitable for numerical evaluation, is given here, equation 29.