Zeros of Polynomials in Derivatives of Automorphic $L$-functions

Let ${\mathfrak{F}}_m$ be the set of all cuspidal automorphic representations of ${\mathrm{G}\mathrm{L}}_m({\mathbb{A}}_{\mathbb{Q}})$, and let $F(s,\mathit{\pi })$ be a polynomial in the derivatives of $L$-functions associated with representations ${\pi }_u\in {\cup }_{m=1}^{\mathrm{\infty }}{\mathfrak{F}}_m$. We establish an asymptotic formula for the number of nontrivial zeros of $F(s,\mathit{\pi })$ with $0<\mathrm{Im}(s)<T$. We explicitly determine the main term of this formula in terms of the degrees, the ranks, the arithmetic conductors, and the orders of differentiation of the component $L$-functions. Furthermore, we show that, under certain conditions, almost all nontrivial zeros of $F(s,\mathit{\pi })$ lie near the critical line $\mathrm{Re}(s)=1\mathrm{/}2$.