I find a lot of literature describing $\Gamma(n,z)$ for $n$ an integer, but in this case, I care about the $n$ being complex and $z$ being fixed to $1$. According to wikipedia, it appears that $\Gamma(s,z)$ is known to be entire for fixed $z\not=0$ (in particular, $z=1$). My main question is if all the zeros are isolated to a half plane, specifically is $$\Gamma(z,1)\not=0\ \ \ \ \text{for}\ \ \ \ \Re(z)>0$$ or $$\Gamma(z,1)\not=0\ \ \ \ \text{for}\ \ \ \ \Re(z)<0$$ I believe this is this the case but could use a reference or a proof.
UPDATE:
According to wikipedia, it appears that Gauss's continued fraction is $$e\cdot\Gamma(z,1)=\frac{1}{1+\frac{1-z}{1+\frac{1}{1+\frac{2-z}{1+\frac{1}{1+\frac{3-z}{\ddots}}}}}}$$ which is said to converge uniformly on compact subsets that don't contain poles/zeros of the fractional functions.
If this is the case, it appears to suggest that $\Gamma(z,1)^{-1}$ is also an analytic everywhere, except possibly the positive integers (where the continued fraction does not hold). This suggests that $$\Gamma(z,1)\not= 0$$ for all $\Re(z)<0$.
Does this appear to be a valid argument?
UPDATE:
I think I misread, it seems it only the poles of the covergent that yield a problem for a continued fraction's convergence. If so, this would suggest that $\Gamma(z,1)\not=0$ for all $z$ and that $\Gamma(z,1)^{-1}$ is also an entire function.