Let $\alpha \in (0,2)$ and let $\eta(t,x)$ be the density of alpha stable subordinator, that is $$ \int_0^{\infty}\eta(s,x)e^{-xt} dx = e^{-st^{\alpha/2}}, ~s\in(0,\infty).$$ I would like to calculate for $0< \gamma < \frac{1}{2}$, $$ \int_0^{\infty}s^{\gamma}\eta(s,1)ds. $$
Is there any article or book where integral of such function (or similar to this one) is calculated ?