  We propose non-asymptotic controls of the cumulative distribution function
$P(|X_{t}|\ge \varepsilon)$, for any $t>0$, $\varepsilon>0$ and any L\'evy
process $X$ such that its L\'evy density is bounded from above by the density
of an $\alpha$-stable type L\'evy process in a neighborhood of the origin. The
results presented are non-asymptotic and optimal, they apply to a large class
of L\'evy processes.
