Zero distribution of finite order Bank--Laine functions

It is known that a Bank-Laine function $E$ is a product of two normalized solutions of the second order differential equation $f\mathrm{"}+Af=0$ $(†)$, where $A=A(z)$ is an entire function. By using Bergweiler and Eremenko’s method of constructing transcendental entire function $A(z)$ by gluing certain meromorphic functions with infinitely many times, we show that, for each $\lambda \in [1,\mathrm{\infty })$ and each $\delta \in [0,1]$, there exists a Bank–Laine function $E$ such that $E=f_1{f}_2$ with $f_1$ and $f_2$ being two entire functions such that $\lambda (f_1)=\delta \lambda$ and $\lambda (f_2)=\lambda$, respectively. We actually provide a simpler construction of the special Bank–Laine functions given by Bergweiler and Eremenko.