  We evaluate the first three moments of central values of a family of qudratic
Hecke $L$-functions in the Gaussian field with power saving error terms. In
particular, we obtain asymptotic formulas for the first two moments with error
terms of size $O(X^{1/2+\varepsilon})$. We also study the first and second
mollified moments of the same family of $L$-functions to show that at least
$87.5\%$ of the members of this family have non-vanishing central values.
