Hello, I have been working on those Zeta(2n+1) identities that involves infinite series with exp(-Pi*n) lately and I think I have found new formulas. They are interesting since they converge fairly rapidly, for example, that one for Zeta(5) : /infinity \ 5 | ----- | 694 Pi 6280 | \ 1 | Zeta(5) - ------- + ---- | ) --------------------| 204813 3251 | / 5 | | ----- n (exp(4 Pi n) - 1)| \ n = 1 / /infinity \ | ----- | 296 | \ 1 | - ---- | ) --------------------| 3251 | / 5 | | ----- n (exp(5 Pi n) - 1)| \ n = 1 / /infinity \ | ----- | 1073 | \ 1 | + ---- | ) ---------------------| 6502 | / 5 | | ----- n (exp(10 Pi n) - 1)| \ n = 1 / /infinity \ | ----- | 37 | \ 1 | - ---- | ) ---------------------| 6502 | / 5 | | ----- n (exp(20 Pi n) - 1)| \ n = 1 / It converges at a rate of 5.45 decimal digits per iteration. lprint(%): Zeta(5)-694/204813*Pi^5+6280/3251*sum(1/n^5/(exp(4*Pi*n)-1),n = 1 .. infinity)-\ 296/3251*sum(1/n^5/(exp(5*Pi*n)-1),n = 1 .. infinity)+1073/6502*sum(1/n^5/(exp( 10*Pi*n)-1),n = 1 .. infinity)-37/6502*sum(1/n^5/(exp(20*Pi*n)-1),n = 1 .. infinity): I have similar identities for Zeta(9) and Zeta(13) but not yet Zeta(3). As you can see there is some general patterns, other formulas can be found here : http://www.lacim.uqam.ca/~plouffe/ Simon Plouffe