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. $E[ X^{1/2}]$.