# where the exponential and the gamma function are integral multiples of one another

Let $g(z) = \sin\left(\frac{\pi e^{z}}{\Gamma(z)}\right)\sin\left(\frac{\pi \Gamma(z)}{e^{z}}\right)$. $g(z)$ has a root in $\mathbb{C}$ (domain colored below) whenever there exists $n\in\mathbb{Z}$ such that $ne^{z}=\Gamma(z)$ or $n\Gamma(z)=e^{z}$

Do you see the double root at $z\approx 8.1636558707955126\ldots$ or so? That’s because at that point $2\exp(z)=\Gamma(z)$. There’s a whole necklace of double roots on the right side of the picture.