Is there any "easy" way to calculate fractional moments from Laplace transform. To be more specyfic let us consider the following example. Let $X$ be a positive random variable and $L(\theta) := E \exp (-\theta X)$ be its Laplace transform. Of course it is easy to calculate $E X^n$ where $n$ is a natural number but what with e.g. $E X^{1/2}$.