I'm interested in the zeros of the lower incomplete gamma function $$\gamma(s,x) = \int_0^x t^{s-1}e^{-t}\,dt \,.$$ In http://www.jstor.org/stable/2007135, Franklin (1919) derives some numerical approximations for the zeros of $\gamma(s,1)$ in terms of $s$, however I've been unable to find any kind of closed form solution or more modern description of these zeros. Does anyone know of any references that discuss the zeros of the lower incomplete gamma function, either in general or at $x=1$?