Characterize the zeros of the deformation factor

Characterize the zeros of the deformation factor \(M_a(s)\) for general positive real parameters \(a\), including their locations and distribution in the complex plane.

Background

The deformation factor is defined by $$M_a(s)=\frac{1}{\Gamma(a)\Gamma(s)}\int_0^\infty u^{s-1}e^{-u}\gamma(a,u)\,du$$, and the paper states that it is entire in $s$, has no poles, and possesses isolated zeros. These zeros generate additional zeros of $k_a(s)$ that need not arise from zeros of the Riemann zeta function.

For $a=1$, the factor has a special relation to the alternating zeta function, but no general closed-form description of its zeros is available for arbitrary $a>0$. The paper mentions numerical quadrature, series evaluation, and root-finding as possible tools for studying them.