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We evaluate the first three moments of central values of a family of qudraticHecke $L$-functions in the Gaussian field with power saving error terms. Inparticular, we obtain asymptotic formulas for the first two moments with errorterms of size $O(X^{1/2+\varepsilon})$. We also study the first and secondmollified 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.

Moments and Non-vanishing of central values of Quadratic Hecke $L$-functions in the Gaussian Field