Determine the full analyticity and zero-free domains of the deformed zeta kernel

Determine the full extent of the analyticity and zero-free domains of the variant fractional gamma-regularized zeta kernel k_a(s), particularly as the parameter a varies.

Background

The paper defines $k_a(s)$ as a deformation of the Riemann zeta function satisfying $k_a(s)=\zeta(s)M_a(s)$, where $M_a(s)$ is a gamma-based deformation factor. The paper establishes convergence and analyticity in the half-plane $\Re(s)>1$, and observes that the zero structure of $k_a(s)$ depends both on the zeros of $\zeta(s)$ and on any additional zeros of $M_a(s)$.

Although the paper notes that the analyticity and zero-free regions of $k_a(s)$ are at least as broad as those of $\zeta(s)$ in $\Re(s)>1$, it does not determine their complete extent, especially for varying positive values of $a$.