What is the Laplace transform of gamma function?

Question

What is $$\mathfrak L (\Gamma(z))$$? And how can it be derived?

$$\Gamma(z)=\int_0^\infty t^{z-1}e^t dt$$

$$\mathfrak L(\Gamma(z)) = \int_0^\infty \int_0^\infty t^{z-1}e^t dt e^{-sz} dz$$

Yes, of course I tried to find on google but I could only find something like laplace transform using the gamma function, and inverse Laplace transform of gamma function, etc..

Also I tried by myself, but no fruit..

This result may be applied to solving gamma functional equation, which is in another question.

Answer 1

You can't find it because it does not exist: $$\lim_{n \to \infty} \frac{n!}{e^{an}}=\infty \quad \forall a \in \mathbb{R}$$