$\lim _{N\to \infty } Q^{-1}(\frac{N}{2},0.5)= ? $

Question

What is $\lim _{N\to \infty } Q^{-1}(\frac{N}{2},0.5)= ? $ for $N>0$

where: $Q^{-1}(s,z)$ is the "Inverse of regularized incomplete gamma function" for $x$ in $z=Q(s,x)$.

$Q(s,x)=\frac{\Gamma(s,x)}{\Gamma(s)}$ is the upper incomplete gamma function; where $\Gamma(s,x)=\int_x^\infty t^{s-1}e^{-t}dt$ is regularized by gamma function $\Gamma(s)$.

Thanks