Non-vanishing of Cubic Dirichlet L-Functions over the Eisenstein Field

We establish an asymptotic formula for the first moment and derive an upper bound for the second moment of L-functions associated with the complete family of primitive cubic Dirichlet characters defined over the Eisenstein field. Our results are unconditional, and indicate that there are infinitely many characters within this family for which the L-function $L(s,\chi )$ does not vanish at the central point $s=1\mathrm{/}2$.