Zeros of polynomials of derivatives of zeta functions

Let $P_s\in {\mathcal{D}}_s[X_0,X_1,\dots ,X_l]$ be a polynomial whose coefficients are the ring of all general Dirichlet series which converge absolutely in the half-plane $\mathrm{\Re }(s)>1\mathrm{/}2$. In the present paper, we show that the function $P_s(L(s),L^{(1)}(s),\dots ,L^{(l)}(s))$ has infinitely many zeros in the vertical strip $D:= { s \in {$\mathbb{C}$} : 1/2 < \Re (s) <1}$if$L(s)$ishybridlyuniversaland$P_s \in $\mathcal{D}$_s[X_0,X_1, \ldots,X_l]$isapolynomialsuchthatatleastoneofthedegreeof$X_1,\ldots,X_l$ is greater than zero. As a corollary, we prove that the function $(d^k\mathrm{/}d{s}^k)P_s(L(s))$ with $k\in \mathbb{N}$ has infinitely many zeros in the strip $D$ when $L(s)$ is hybridly universal and $P_s \in $\mathcal{D}$_s[X]$ is a polynomial with degree greater than zero. The upper bounds for the numbers of zeros of $P_s(L(s),L^{(1)}(s),\dots ,L^{(l)}(s))$ and $(d^k\mathrm{/}d{s}^k)P_s(L(s))$ are studied as well.