For a (non-negative) random variable $X,$ define its Laplace transform as $$\mathbb{E}(e^{-\lambda X}) =: L(\lambda),$$ for $\lambda > 0.$ We know that this uniquely determines the law of $X$ as long as $X \geq 0.$ Furthermore, by taking series expansions of the exponential function, we can calculate moments of $X$ from the derivatives of $L$ and evaluating these at $0.$
What property of $L$ will give us the maximum $p$ so that $X \in L^p(\mathbb{P})?$