First Moment of Quadratic Hecke $L$-Functions with Lower Order Term

We evaluate the first moment of the family of primitive quadratic Hecke $L$-functions in the Gaussian field using the method of double Dirichlet series under the Riemann hypothesis and the Lindelöf hypothesis. We obtain asymptotic formulas with secondary main terms and error terms of size that is one quarter of that of the main term.