On distribution of zeros of generalized functions of Mittag-Leffler's type

The following two problems related to the entire function $$ \Phi_{\rho_1, \rho_2}(z, \mu_1, \mu_2)=\sum_{n=0}^{\infty} \frac {z^n}{\Gamma (\mu_1 +n/{\rho_1}) \Gamma (\mu_2 +n/{\rho_2})} \;, \quad \rho_1, \rho_2, \mu_1, \mu_2 > 0 , $$ are considered.
1. For $0 < \rho_1 \leq 1/2$, determine the values of $\mu_1, \rho_2, \mu_2$ such that all zeros of $\Phi_{\rho_1, \rho_2}(z, \mu_1, \mu_2)$ are negative.
2. For $1/2 < \rho_1 < 1$, determine the values of $\mu_1, \rho_2, \mu_2$ such that all zeros of $\Phi_{\rho_1, \rho_2}(z, \mu_1, \mu_2)$ are situated outside the angle $\{z : |\arg z| \leq \pi/(2 \rho_1) \}$. The results point out several values of $\mu_1$ and all values of $\rho_2 > 0, \mu_2 > 0$ with the above properties. Moreover, we obtain some results on $a$-points of the function $\Phi_{\rho_1, \rho_2}(z, \mu_1, \mu_2)$.