Lattice points on the trace 1

Given a rectangle $ABCD$ with side $AB$ = $CD$ = $a$ and side $BC$ = $AD$ = $b$. A curve is traced by point $D$ of the rectangle as point $A$ slides along the x-axis and point $B$ slides along the y-axis.

The curve with $a=2$ and $b=4$ is shown below.



There are 4 lattice points $(-4,0),(-2,-4),(4,0),(2,4)$ on the curve above. Let $C(a,b)$ be the number of distinct lattice points on the curve with the rectangle side $a$ and $b$. For example, $C(2,4)=4$, $C(5,5)=12$ and $C(6,3)=4$.

Define $S(n)={\sum }_{a=1}^n{\sum }_{b=1}^{n}C(a,b)$. You are given $S(10)=432$, $S(50)=10960$ and $S(100)=44128$.

Find $S({10}^8)$.