Here's another: DedekindEta[I/(2 Sqrt[5])] == (1/(2 \[Pi]^(3/4)) Sqrt[Gamma[1/20] Gamma[9/20]] (1/10 (5 - Sqrt[5]) (1 - InverseEllipticNomeQ[E^(-(\[Pi]/Sqrt[5]))]))^(1/8))== Sqrt[(1/(2^(3/4) 5^(1/8)) - ( 2^(3/4) 5^(5/8))/(5 + Sqrt[5])^(3/2)) Gamma[1/20] Gamma[9/20]]/( 2 \[Pi]^(3/4)) thanks to the unlikelihood -11 + 5 Sqrt[5] == (1600 Sqrt[5])/(5 + Sqrt[5])^5 Corey's denester reduced the 8th root of a nested radical to the square root of a radical of the same depth. On Tue, Aug 11, 2015 at 7:30 AM, Bill Gosper <billgosper@gmail.com> wrote:
"The Chowla–Selberg formula gives a formula for a finite product of values of the eta functions. By combining this with the theory of complex multiplication <https://en.wikipedia.org/wiki/Complex_multiplication>, one can give a formula for the individual absolute values of the eta function as [image: \Im(\tau)|\eta(\tau)|^4 = \frac{\alpha}{4\pi\sqrt{|D|}} \prod_r\Gamma(r/|D|)^{\chi(r)\frac{w}{2h}}]
for some algebraic number α."
where chi(r) = JacobiSymbol[r,D] and " *−D* is the discriminant <https://en.wikipedia.org/wiki/Discriminant> of an imaginary quadratic field <https://en.wikipedia.org/wiki/Quadratic_field>".
That algebraic number can often be expressed in terms of ModularLambda[tau]:
DedekindEta[(I Sqrt[11])/2] == 1/(2 \[Pi]^(7/8)) (1/ 11 Gamma[1/22] Gamma[3/22] Gamma[5/22] Gamma[9/22] Gamma[15/ 22])^(1/4) (1 - ModularLambda[I Sqrt[11]])^(1/8)
\[Eta][E^(-Sqrt[3] \[Pi])] == ( Gamma[1/3]^(3/2) (3 (1 - InverseEllipticNomeQ[E^(-Sqrt[3] \[Pi])]))^(1/8))/(2 \[Pi])
\[Eta][E^(-Sqrt[5] \[Pi])] == 1/(2 5^(3/8) \[Pi]^(3/4)) Sqrt[Gamma[1/20] Gamma[9/20]] (1/ 2 (5 - Sqrt[5]) (1 - InverseEllipticNomeQ[E^(-Sqrt[5] \[Pi])]))^( 1/8)
\[Eta][E^(-(\[Pi]/Sqrt[5]))] == (1/(2 \[Pi]^(3/4))) Sqrt[Gamma[1/20] Gamma[9/20]] (1/ 10 (5 - Sqrt[5]) (1 - InverseEllipticNomeQ[E^(-(\[Pi]/Sqrt[5]))]))^(1/8)
It's probably not cheating because it doesn't always work.
( ModularLambda[r] = InverseEllipticNomeQ[E^(I*PI*r)])
--rwg