Stirling's approximation of a Laplace transform

Question

I have got the Laplace transform as follows:

$$L_X(s) = \exp(-s)\sum_{t = 0}^{\infty} \frac{s^t}{t!}\exp(-Ct^{2/\alpha}),$$ where $C$ and $\alpha$ are positive constants.

I want to prove that, using Stirling's approximation, we have

$$-\log(L_X(s)) \sim Cs^{2/\alpha}$$ as $s \to \infty$.