I'm trying to calculate the Laplace exponent of a standar $\alpha-$stable subordinator.
An $\alpha-$stable subordinator has Lévy measure $\frac{c}{x^{1+\alpha}}dx,$ where $\alpha\in(0,1)$ and $c$ is a positive constant.
Now, if the subordinator is $\alpha-$stable then has drift zero. So Laplace exponent has the form $$\Phi(\lambda)=c\int_{0}^{\infty}(1-e^{-\lambda x})\frac{dx}{x^{1+\alpha}}.$$
I've read on a book that Laplace exponent of this kind of subordinators have the form $$\Phi(\lambda)=\frac{\alpha}{\Gamma(1-\alpha)}\int_{0}^{\infty}(1-e^{-\lambda x})\frac{dx}{x^{1+\alpha}}.$$
I've proved that such measure is Lévy measure; the hypotesis $\alpha\in(0,1)$ is indispensable to have finite integrals independent of the constant $c.$ So I'd like to prove that constant $c$ is equal to $\frac{\alpha}{\Gamma(1-\alpha)}$ but I don't get it.I was trying to use Gamma function but I don't find something useful.
Any kind of help is thanked in advanced.