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 <A HREF="http://web.ist.utl.pt/berberan/data/95.pdf">here</A>, equation 29.