Does $\sum_{k\geq0}\frac{z^k}{k!^s}$ only have negative real 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 following function given by the infinite series,$$\sum_{k\geq0}\frac{z^k}{k!^2}$$has infinitely many real zeros only (from Bessel function theory).

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