  Let $X$ be a real valued L\'evy process that is in the domain of attraction
of a stable law without centering with norming function $c.$ As an analogue of
the random walk results in \cite{vw} and \cite{rad} we study the local
behaviour of the distribution of the lifetime $\zeta$ under the characteristic
measure $\underline{n}$ of excursions away from 0 of the process $X$ reflected
in its past infimum, and of the first passage time of $X$ below $0,$
$T_{0}=\inf \{t>0:X_{t}<0\},$ under $\mathbb{P}_{x}(\cdot),$ for $x>0,$ in two
different regimes for $x,$ viz. $x=o(c(\cdot))$ and $x>D c(\cdot),$ for some
$D>0.$ We sharpen our estimates by distinguishing between two types of path
behaviour, viz. continuous passage at $T_{0}$ and discontinuous passage. In the
way to prove our main results we establish some sharp local estimates for the
entrance law of the excursion process associated to $X$ reflected in its past
infimum.
