  We prove, assuming the generalized Riemann Hypothesis (GRH) that there is a
positive density of $L$-functions associated with primitive cubic Dirichlet
characters over the Eisenstein field that do not vanish at the central point
$s=1/2$. This is achieved by computing the first mollified moment, which is
obtained unconditionally, and finding a sharp upper bound for the higher
mollified moments for these $L$-functions, under GRH. The proportion of
non-vanishing is explicit, but extremely small.
