  We find an expression for the joint Laplace transform of the law of
$(T_{[x,+\infty[},X_{T_{[x,+\infty[}})$ for a L\'evy process $X$, where
$T_{[x,+\infty[}$ is the first hitting time of $[x,+\infty[$ by $X$. When $X$
is an $\alpha$-stable L\'evy process, with $1<\alpha<2$, we show how to recover
from this formula the law of $X_{T_{[x,+\infty[}}$; this result was already
obtained by D. Ray, in the symmetric case and by N. Bingham, in the case when
$X$ is non spectrally negative. Then, we study the behaviour of the time of
first passage $T_{[x,+\infty[},$ conditioned to $\{X_{T_{[x,+\infty[}} -x \leq
h\}$ when $h$ tends to $0$. This study brings forward an asymptotic variable
$T_x^0$, which seems to be related to the absolute continuity of the law of the
supremum of $X$.
