The classical mean value result of Murty and Murty (1991) and Bump, Friedberg, and Hoffstein (1990) on derivatives of modular form L-functions $L(s,f)$ proves (roughly speaking) the existence of infinitely many imaginary quadratic fields $K$ for which the Heegner hypothesis holds, ie., every prime dividing the level of a given modular form $f$ splits in $K$, such that a $L(s,f)$ has a simple pole at $s=k$, with $2k$ being weight of $f$, and such that $L(k,f,\chi)= 0$ for the twist $\chi$ associated to $K$ (or vice versa).

I am looking to see if this result has been generalized to (any) fields $K$ in which the Heegner hypothesis fails to hold. Searching through the literature has not proved to be a simple task.

The classical mean value result of Murty and Murty (1991) and Bump, Friedberg, and Hoffstein (1990) on derivatives of modular form L-functions $L(s,f)$ proves (roughly speaking) the existence of infinitely many imaginary quadratic fields $K$ for which the Heegner hypothesis holds, ie., every prime dividing the level of a given modular form $f$ splits in $K$, such that a $L(s,f)$ has a simple zero at $s=k$, with $2k$ being weight of $f$, and such that $L(k,f,\chi)\neq 0$ for the twist $\chi$ associated to $K$ (or vice versa).

I am looking to see if this result has been generalized to (any) fields $K$ in which the Heegner hypothesis fails to hold. Searching through the literature has not proved to be a simple task.