On distribution of zeros for a class of meromorphic functions

In this article some class $\mathcal{K}_0$ of meromorphic functions is introduced. Each function $y(z)$ from $\mathcal{K}_0$ satisfies the functional equation $y(z)=b_y(z)y(1-z)$ with its own "Riemann's multiplier" $b_y(z)$ which is a meromorphic function with real zeros and poles. All poles of an arbitrary function from $\mathcal{K}_0$ are real and belong to the interval $(\frac12,\frac12+h_1]$, $h_1=h_1(y)$. Using the theory of residues we prove some relation connecting the following magnitudes: $\mathcal{P}_y$, the sum of all orders of poles of $y \in \mathcal{K}_0$; $\mathcal{N}_y(T)$, the sum of multiplicities of all zeros of $y$ having the form $\frac12 +i\tau$, $|\tau|