E.g., EllipticK[1/8 (1 + Sqrt[3])^2] == (3^(3/4) Gamma[1/3]^3)/(4 2^(1/3) π) EllipticTheta[4, 0, E^(-Sqrt[2] \[Pi])] == Sqrt[Gamma[1/8] Gamma[3/8]]/(2^(7/8) \[Pi]^(3/4)) KleinInvariantJ[2 I] == 1331/8 KleinInvariantJ[69^105+2 I] == 1331/8 (Well, whaddya expect from an invariant?) For years I've griped about their absence from Mathematica and from tables (Dlmf, functions.wolfram.com, Mathworld, Wikipedia,...) This is hard to excuse, since such values are nearly all expressible in terms of DedekindEta. E.g., EllipticK[(16 DedekindEta[\[Tau]]^8 DedekindEta[4 \[Tau]]^16)/DedekindEta[2 \[Tau]]^24] == (\[Pi] DedekindEta[2 \[Tau]]^10)/(2 DedekindEta[\[Tau]]^4 DedekindEta[4 \[Tau]]^4) so we'd be well off just mastering DedekindEta[algebraic (north of 0)]. But the biggest collection of values I've ever seen (besides my own) is the *four* in en.wikipedia.org/wiki/Dedekind_eta_function#Special_values . Yet there are papers claiming closed forms for a huge fraction of cases (too many to tabulate). One such is http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.63.5854&rep=rep1&ty... which claims to have extended coverage to the likes of DedekindEta[Sqrt[-11]] and DedekindEta[(1 + Sqrt[-11])/3], or at least their Abs's. After several pages of dense hair, they give a formula for a product of these. Then more hair. Then the quotient. In terms of the "Tribonacci constant"! " thus completing our eta evaluation." Which they don't bother to spell out. Lucky for them. Both the product and quotient formulas are WROMG! Digging through my old Macsyma files, DedekindEta[ Sqrt[11]*I] == ((3^(1/8)*(Gamma[1/22])^(1/4)*(Gamma[3/22])^(1/ 4)*(Gamma[5/22])^(1/4)*(Gamma[9/22])^(1/4)*(Gamma[15/22])^(1/ 4))/(2*22^(1/4)*(2^(1/3)*(21*Sqrt[33] + 283)^(1/3) + 2^(1/3)*(283 - 21*Sqrt[33])^(1/3) + 8)^(1/8)*Pi^(7/8))) where the radical simplifies, and is currently choking Corey's denester. But my result is merely empirical. The paper's result is supported by hairy theory. And misprinted, at best. Given the Logs of all its factors, FindIntegerNullVector (PSLQ) finds DedekindEta[ I Sqrt[11]] == ((Gamma[1/11] Gamma[3/11] Gamma[4/11] Gamma[5/11] Gamma[9/11])^(1/5) Root[-1 + 2 #1 - 2 #1^2 + 2 #1^3 &, 1])/ (11^(11/40) (Gamma[2/11] Gamma[6/11] Gamma[7/11] Gamma[8/11] Gamma[10/11])^(3/10)) Modest proposal: We at least have a structure theorem. Or a darn good conjecture. Just evaluate DedekindEta by automated PSLQ|RootApproximant. This could even be made rigorous if there's a theorem stating how close you can get without being exact. Someday, PSLQ will seem a one-button operation, like continued fraction. --rwg MathIsFun for you-know-who.