What is the probability that X takes even values?

Am trying to show that the probability a count r.v $X$ takes even values is given by

$\frac 1 2( 1 + G(-1)),$ where $G(t)$ is its probability generating function.

I know that due to possible symmetry then the $P(\text{odd}) + P(\text{even}) =1.$

Kindly help any initial stages.