Zero of non analytic function

Question

Let a function $L=L(z)$ be analytic, for $\mathrm{Re}\, z>0$, and be singular at $0$, however, $L(0)=c$ be finite. Let also $L'(0)$ be finite as well, however, $L'(0)\neq 0$. For example, $$ L(z)=\int_0^\infty \frac{e^{-z t}}{1+t^3}dt. $$ May I state that $$ \frac{1}{L(z)-c}=\frac{A}{z}+M(z), $$ where $A$ is a constant and $M(z)$ is a function analytic on $\mathrm{Re}\,z>0$, singular at $z=0$ and also with finite $M(0)$?