How to solve $x!=5^x$?

Question

Or, more generally, $$\Gamma (x+1)=\int_0^{\infty}t^{x}e^{-t}dt=p^x$$ with $p \in \mathbb{Z}^+$ and $x \in \mathbb{C}$.

Perhaps begin with $\large p^x=p^x \lim_{n \rightarrow \infty}[e^{-0}-e^{-n}]=\int_0^{\infty}p^xe^{-t}dt$.

Then

$$\int_0^{\infty}(t^{x}-p^x)e^{-t}dt=0$$

which I'm not sure how to solve. It smells of integration by parts, but I can't see anything feasible given that $x$ isn't a constant.

Answer 1

Here's a plot of the complex roots of the equation $\Gamma(z+1) = 5^z$ near the origin.

Numerically it appears that the only other roots are those continuing along the three "rays" of roots pictured: the ray along the negative real axis (here the roots are near-integers) and the two symmetric arms which appear to tend to $\pi/3$ and $-\pi/3$ radians.