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.