  This work derives 5 methods to evaluate families of odd zeta values by
combining a power of $\pi$ with Lambert series whose ratios of successive terms
tend to $e^{-\pi\sqrt{a}}$ with integers $a\ge7$, outperforming Ramanujan's
results with merit $a=4$. Families with $a=7$ and $a=8$ evaluate $\zeta(2n+1)$.
Families with $a=9$ and $a=16$ evaluate $\zeta(4n+1)$ with faster convergence.
A fifth family with $a=12$ evaluates $\zeta(6n+1)$ and gives the fastest
convergence for $\zeta(6n+7)$. Members of three of the families were discovered
empirically by Simon Plouffe. An intensive new search strongly suggests that
there are no more than 5 families with integers $a\ge7$. There are at least 20
families that involve Lambert series with rational $a>4$. Quasi-modular
transformations of Lambert series resolve rational sequences that were
discovered empirically. Expansions of Lambert series in polylogarithms,
familiar from quantum field theory, provide proofs of all known evaluations
with rational merit $a>4$.
