Frobenius & Special Functions · Superchallenge problem

Let $j_1 < j_2 < j_3 < \cdots$ be the positive zeros of the Bessel function $J_1(x) = \displaystyle\sum_{k\ge0}\frac{(-1)^k}{k!\,(k+1)!}\left(\frac x2\right)^{2k+1}$. Given that $\dfrac{2J_1(x)}{x} = \displaystyle\prod_{k\ge1}\left(1 - \frac{x^2}{j_k^2}\right)$, compute $\displaystyle\sum_{k\ge1}\frac{1}{j_k^2}$. Express your answer as a common fraction.