A Necessary Condition for the Completeness of the System $\{e^{-\lambda_nt}\mid\operatorname{Re}\lambda_n>0\}$ in the Spaces $C_0(\mathbb R_+)$ and $L^p(\mathbb R_+)$, $p>2$

We obtain a necessary condition for the completeness of the system
$$ e(\Lambda)=\{e^{-\lambda_nt}\mid\operatorname{Re}\lambda_n>0,\,n\in\mathbb Z\} $$
in the spaces $C_0$ and $L^p(\mathbb R_+)$, $p>2$, for the case in which the set of limit points of the sequence $\{\lambda_n\}$ is countable and separable.