  We evaluate asymptotically the smoothed first moment of central values of
families of quadratic, cubic, quartic and sextic Hecke $L$-functions over
various imaginary quadratic number fields of class number one, using the method
of double Dirichlet series. In particular, we obtain asymptotic formulas for
the quadratic families with error terms of size $O(X^{1/4+\varepsilon})$ under
the generalized Riemann hypothesis.
