Let $$ f_N(z) = \sum_{n=N}^\infty \frac{z^n}{n!},$$ where $N$ is a positive integer.

Where are the (complex) zeros of those functions located?

It would be sufficient for me to know what the smallest (in absolute value) non-zero zeros are for a general $N$. Maybe an approximation of those zeros would also help.

Up to now I just found something about the zeros of the polynomial $e(z) - f_N(z)$ (i.e. the truncated taylor expansion of $e(z)$). The keyword here is "Szegö curve".