  We consider the standard model of first-passage percolation on $\mathbb{Z}^d$
($d\geq 2$), with i.i.d. passage times associated with either the edges or the
vertices of the graph. We focus on the particular case where the distribution
of the passage times is the Bernoulli distribution with parameter $1-\epsilon$.
These passage times induce a random pseudo-metric $T_\epsilon$ on
$\mathbb{R}^d$. By subadditive arguments, it is well known that for any
$z\in\mathbb{R}^d\setminus \{0\}$, the sequence $T_\epsilon (0,\lfloor nz
\rfloor) / n$ converges a.s. towards a constant $\mu_\epsilon (z)$ called the
time constant. We investigate the behavior of $\epsilon \mapsto \mu_\epsilon
(z)$ near $0$, and prove that $\mu_\epsilon (z) = \| z\|_1 - C (z) \epsilon
^{1/d_1(z)} + o ( \epsilon ^{1/d_1(z)}) $, where $d_1(z)$ is the number of non
null coordinates of $z$, and $C(z)$ is a constant whose dependence on $z$ is
partially explicit.
