  We determine ${\bar{\rm{lcr}}}(K_n)$, the rectilinear local crossing number
of the complete graph $K_n$ for every $n$. More precisely, for every $n \notin
\{8, 14 \}, $ \[ {\bar{\rm{lcr}}}(K_n)=\left\lceil \frac{1}{2} \left(
n-3-\left\lceil \frac{n-3}{3} \right\rceil \right) \left\lceil \frac{n-3}{3}
\right\rceil \right\rceil, \] ${\bar{\rm{lcr}}}(K_8)=4$, and
${\bar{\rm{lcr}}}(K_{14})=15$.
