The rectilinear local crossing number of $K_n$

We determine $\overline{\mathrm{l}\mathrm{c}\mathrm{r}}(K_n)$, the rectilinear local crossing number of the complete graph $K_n$ for every $n$. More precisely, for every $n \notin {8, 14 }, $ $$\overline{\mathrm{l}\mathrm{c}\mathrm{r}}(K_n)=\lceil \frac{1}{2}(n-3-\lceil \frac{n-3}{3}\rceil )\lceil \frac{n-3}{3}\rceil \rceil ,$$ $\overline{\mathrm{l}\mathrm{c}\mathrm{r}}(K_8)=4$, and $\overline{\mathrm{l}\mathrm{c}\mathrm{r}}(K_{14})=15$.