Let q=e2πiτ. Given τ=√−n or τ=1+√−n2, then the solutions α,β,γ to the following equations,
2F1(12,12;1;1−α)2F1(12,12;1;α)i=√4τ2F1(13,23;1;1−β)2F1(13,23;1;β)i=√3τ2F1(14,34;1;1−γ)2F1(14,34;1;γ)i=√2τ are given by, α=16u81+16=(√2u2)8=(ϑ2(q)ϑ3(q))4β=27v121+27=(3v32+3)3=(c(q)a(q))3γ=64w241+64=(8w82+8)2=(f(q)d(q))2 where, u1=η(τ)η(4τ),u2=η3(2τ)η(τ)η2(4τ)v1=η(τ)η(3τ),v2=η(τ/3)η(3τ)w1=η(τ)η(2τ),w2=η(τ/2)η(2τ) and where the functions of q=e2πiτ are the Jacobi and Borwein theta functions discussed in Entry 38. Also, α=λ(2τ) with the modular lambda function λ(τ).