Special squarefree sum

A squarefree number is called special if it can be expressed as the sum of two positive squares. For example, $65(=1^2+8^2)$ is a special squarefree number while $15$ is not.

Let $S(N)$ be the sum of all special squarefree numbers from $1$ to $N$. You are given that $S(10)=17$, $S(1000)=81665$, $S({10}^6)=61081126084$.

Find $S({10}^{12})$.