More generally, if $1\le k\le 9$ is an integer,
$$S_k := \sum_{n=0}^\infty\biggl( \frac{1}{(10n-k)^2} + \frac{1}{(10n+k)^2}
   \biggr) = \frac{\pi^2}{100\sin^2(\pi k/10)}.$$
Your sum is $S_1-S_3$.