On a Gross conjecture over imaginary quadratic fields

Let $k$ be an imaginary quadratic number field, and $F\mathrm{/}k$ a finite abelian extension of Galois group $G$. We show that a Gross conjecture concerning the leading terms of Artin $L$-series holds for $F\mathrm{/}k$ and all rational primes which are split in $k$ and which do not divide $6$.