I am struggling with the derivation of the Lévy-measure of a Gamma-process $X_t$ with law $p_t(x)= \frac{\lambda^{ct}}{\Gamma(ct)}x^{ct-1}e^{-\lambda x}1_{\lbrace x>0 \rbrace }$. The paper I am reading says it is shown "easily" via the characteristic function and using the Levy-Khintchine formula:
$\Phi(s)=E[e^{isX_1}] = exp( ias - \frac{1}{2}\sigma^2s^2 + \int_{\mathbb{R}}(1-e^{isx} - is\mathbb{1}_{|x|<1})d\nu(x))$
I computed via the pdf of the Gamma-distribution $\Phi(s) = E[e^{isX_t}]= (1-\frac{is}{\lambda})^{-ct}$
But how is the Lévy-measure then derived to be $\nu(x) = \frac{ce^{-\lambda s}}{x}1_{\lbrace 1 > 0\rbrace}$?