Only real negative roots

It might be complicated to decide whether or not a given entire function has only real zeros. So, I'll focus on special examples. It is known that the Bessel function of order $0$, given by the infinite series,$$\sum_{k\geq0}\frac{z^{2k}}{k!^2}$$has infinitely many real zeros that are negative.

QUESTION. Fix an integer $s\geq3$. Can we say the same about the following entire functions?$$\sum_{k\geq0}\frac{z^{sk}}{k!^s}.$$