What can we say about $S=\sum_{n=1}^{\infty}\frac{1}{n^{2}\operatorname{erf}\left(n\right)}$?

Question

I'm really curious If we can find a closed form for this infinite sum :

$$S=\sum_{n=1}^{\infty}\frac{1}{n^{2}\operatorname{erf}\left(n\right)}$$

As $f(x)=\operatorname{erf}(x)\leq 1$ for $0<x$ and $f(x)$ is increasing we have :

$$\frac{\pi^{2}}{6}<S< \frac{\pi^{2}}{6\operatorname{erf}\left(1\right)}$$

So it converges .

Perhaps we can use the Ramanujan's Master Theorem .

Can we say more and find a decent approximation or at best a closed form ?

Thanks everyone !