Rapidly converging formulae for $\zeta (4k\pm 1)$

We provide rapidly converging formulae for the Riemann zeta function at odd integers using the Lambert series $\mathscr{L}q(s) = \sum{n=1}^\infty n^{s} q^{n}/(1-q^n)$,$s=-(4k\pm 1)$\mathrm{.}Ourmainformulafor$\zeta(4k-1)$ converges at rate of about $e^{-\sqrt{15}\pi }$ per term, and the formula for $\zeta (4k+1)$, at the rate of $e^{-4\pi }$ per term. For example, the first order approximation yields $\zeta(3)\approx\frac{\pi ^3 \sqrt{15}}{100} +e^{-\sqrt{15} \pi }\left[\frac{9}{4}+\frac{4}{\sqrt{15}}\sinh (\frac{\sqrt{15} \pi }{2})\right]$whichhasanerroronlyoforder$10^{-10}$.