Differentiating DedekindEta[-1/τ] == Sqrt[-I*τ] DedekindEta[τ], DedekindEta'[-1/τ]==Sqrt[-I τ] τ (DedekindEta[τ]/2+τ DedekindEta'[τ]) Yet note the symmetry between DedekindEta'[-1/τ] and DedekindEta'[τ] for the usual special τ: DedekindEta'[I/2] == ( I Gamma[1/4] (16 \[Pi]^2 - Gamma[1/4]^4))/(32 2^(7/8) \[Pi]^(11/4)), DedekindEta'[2 I] == ( I Gamma[1/4] (16 \[Pi]^2 + Gamma[1/4]^4))/(256 2^(3/8) \[Pi]^(11/4)) DedekindEta'[2 I/3] == ( I Gamma[1/ 4] (144 (-1 + Sqrt[3]) \[Pi]^2 - (-3 + 2 Sqrt[2] 3^(1/4)) (3 + Sqrt[3]) Gamma[1/4]^4))/( 256 2^(13/24) 3^(7/8) (2 - Sqrt[2] 3^(1/4))^(1/4) (-1 + Sqrt[3])^( 5/12) \[Pi]^(11/4)), DedekindEta'[3 I/2] == ( I Gamma[1/ 4] (144 (-1 + Sqrt[3]) \[Pi]^2 + (-3 + 2 Sqrt[2] 3^(1/4)) (3 + Sqrt[3]) Gamma[1/4]^4))/( 1728 2^(1/24) 3^(3/8) (2 - Sqrt[2] 3^(1/4))^(1/4) (-1 + Sqrt[3])^( 5/12) \[Pi]^(11/4)), DedekindEta'[I Sqrt[3]] == ( I Gamma[1/3]^(3/2) (16 2^(2/3) Sqrt[3] \[Pi]^3 + 3 Gamma[1/3]^6))/ (256 3^(7/8) \[Pi]^4), DedekindEta'[I/Sqrt[3]] == -( I 3^(7/8) Gamma[1/3]^(3/2) (-16 2^(2/3) \[Pi]^3 + Sqrt[3] Gamma[1/3]^6)/ (256 \[Pi]^4)), DedekindEta'[I/Sqrt[7]] == 1/(512 \[Pi]^4) I 7^(1/8) Sqrt[ 1/2 Gamma[1/7] Gamma[2/7] Gamma[4/7]] (64 Sqrt[7] \[Pi]^3 - 9 Gamma[1/7]^2 Gamma[2/7]^2 Gamma[4/7]^2), DedekindEta'[I Sqrt[7]] == 1/(3584 7^(1/8) \[Pi]^4) I Sqrt[ 1/2 Gamma[1/7] Gamma[2/7] Gamma[4/7]] (64 Sqrt[7] \[Pi]^3 + 9 Gamma[1/7]^2 Gamma[2/7]^2 Gamma[4/7]^2)} . . . --rwg