For integers $a$ and $b$, we define $D(a,b)$ as the domain enclosed by the parabola $y=x^2$ and the line $y=a\cdot x+b$:
$D(a,b)=\{(x,y)\mid x^2\le y\le a\cdot x+b\}$.

$L(a,b)$ is defined as the number of lattice points contained in $D(a,b)$.
For example, $L(1,2)=8$ and $L(2,-1)=1$.

We also define $S(N)$ as the sum of $L(a,b)$ for all the pairs $(a,b)$ such that the area of $D(a,b)$ is a rational number and $|a|,|b|\le N$.
We can verify that $S(5)=344$ and $S(100)=26709528$.

Find $S({10}^{12})$. Give your answer mod ${10}^8$.