First moment of central values of quadratic Hecke L -functions in the Gaussian field

We evaluate the smoothed first moment of central values of a family of quadratic Hecke L-functions in the Gaussian field using the method of double Dirichlet series. The asymptotic formula we obtain has an error term of size O(X1/4+) under the generalized Riemann hypothesis. The same approach also allows us to obtain asymptotic formulas for all X, Y for a smoothed double character sum involving â N(m)≤X,N(n)≤Y(m n), where (â n) denotes the quadratic symbol in the Gaussian field.