Let $f(z) = \sum_{n=1}^\infty A(n)n^{\frac{k-1}{2}}e(nz)$ be a modular form of weight $k$ for a half integer $k$. Put $$L(s,f) = \sum_{n=1}^\infty \frac{A(n)}{n^s} $$ to be the $L$-function.

Further assume that $f$ is an eigenfunction of the half integral weight Hecke operators.

Has there been located any zeros of this $L$ function, for any choice of $f$, in the critical strip which are not on the critical line $\Re(s) = \frac 12$.