The order (as an entire function) of $E_{\alpha,1}$ is $\frac{1}{\alpha}$. It so happens that entire functions with non-integer order take all complex values infinitely often. So in general $E_{\alpha,1}$ will have infinitely many zeroes. I don't know offhand about the case $\frac{1}{\alpha}\in \mathbb{N}$ though. In that case you can use the functional equation of $\Gamma$.

EDIT: at the time of this answer the OP did not specify that the zero should be real.

The order (as an entire function) of $E_{\alpha,1}$ is $\frac{1}{\alpha}$. It so happens that entire functions with non-integer order take all complex values infinitely often. So in general $E_{\alpha,1}$ will have infinitely many zeroes. I don't know offhand about the case $\frac{1}{\alpha}\in \mathbb{N}$ though. In that case you can use the functional equation of $\Gamma$.

The order (as an entire function) of $E_{\alpha,1}$ is $\frac{1}{\alpha}$. It so happens that entire functions with non-integer order take all complex values infinitely often. So in general $E_{\alpha,1}$ will have infinitely many zeroes. I don't know about the case $\frac{1}{\alpha}\in \mathbb{N}$ though.

The order (as an entire function) of $E_{\alpha,1}$ is $\frac{1}{\alpha}$. It so happens that entire functions with non-integer order take all complex values infinitely often. So in general $E_{\alpha,1}$ will have infinitely many zeroes. I don't know offhand about the case $\frac{1}{\alpha}\in \mathbb{N}$ though. In that case you can use the functional equation of $\Gamma$.