  We develop a new theory of $L$-series based on replacing Dirichlet characters
mod $N$ by symmetric functions mod $N$ in order to calculate explicitly the
sums of infinite series more closely related to $\zeta(2n+1)$, specifically
$\zeta(3)$, $\zeta(5)$. This generalizes the corresponding theory of sums of
$L$-series associated to Dirichlet characters.
