The previous entries dealt with even class numbers. For odd class number h(−d)=1, there are the nine Heegner numbers d=1,2,3,7,11,19,43,67,163. Given the Kronecker symbol (−dm), we propose
Conjecture. Let d>3 with class number h(−d)=1. Then x below is an algebraic number 1x2d=1K(kd)√2π4√d(d∏m=1[Γ(md)](−dm))1/2specificallyxd=21/4Gn=η2(τ)η(τ2)η(τ) with Ramanujan's G-function. This function xd has been discussed in Entry 159. For d=7, then x7=√2, but for the five d≥11, then xd are the real roots of the following five simple cubics,
x3−2x2+2x−2=0x3−2x−2=0x3−2x2−2=0x3−2x2−2x−2=0x3−6x2+4x−2=0 also discussed in Entry 160.
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