Probability of rational values from a mixture of normals

Let $X$ be a random variable with the moment generating function

$$M_X(t)=\frac6{\pi^2}\sum_{n\ge1}\frac{e^{t^2/2n}}{n^2},\quad t\in\mathbb R.$$

Then $P(X\in\mathbb Q)$, where $\mathbb Q$ is the set of rational numbers, equals