[math-fun] 3 was not Dedekind's lucky number

DedekindEta[1/2 + I] == ((-1)^(1/24) 2^(11/16) (1/4)!)/((-1 + Sqrt[2])^(1/4) \[Pi]^(3/4)) == (2^(5/16) ((2 + Sqrt[2]) (I + Sqrt[3]))^(1/4) (1/4)!)/\[Pi]^(3/4) but DedekindEta[1/3 + I] == (1/(\[Pi]^(3/4))) 2 (1/81 (15 + 8 Sqrt[3]) + (2 I ((6 + 7 I) + (7 + 2 I) Sqrt[3]))/( 9 (611 + 344 Sqrt[3] + 77 Sqrt[2 (98 + 55 Sqrt[3])])^(1/3)) + 2/81 (1 + I Sqrt[3]) (611 + 344 Sqrt[3] + 77 Sqrt[2 (98 + 55 Sqrt[3])])^(1/3))^(1/12) (1/4)! Do these (often dreadful) algebraic factors always come out in radicals? —rwg