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We develope the $L$-functions ratios conjecture with one shift in thenumerator and denominator in certain ranges for the family of quadratic Hecke$L$-functions in the Gaussian field using multiple Dirichlet series under thegeneralized Riemann hypothesis. We also obtain an asymptotical formula for thefirst moment of central values of the same family of $L$-functions, obtainingan error term of size $O(X^{1/2+\varepsilon})$.

Ratios conjecture for quadratic Hecke $L$-functions in the Gaussian field