To summarize, given fundamental discriminant d, class number h(−d)=n, complete elliptic integral of the first kind K(kp), and Kronecker symbol (−dm).
Conjecture 1. Let even d=4p for prime p≡1mod4 with even class number h(−d)=n and x=1K(kd/4)√2π2√d(d∏m=1[Γ(md)](−dm))1/(2n)
Conjecture 2. Let odd d=p for prime p≡3mod4 with odd class number h(−d)=n and y=1K(kd)√2π2√d(d∏m=1[Γ(md)](−dm))1/(2n)
then (x,y) are algebraic numbers as seen in entries 160−163 and 165−168. They seem to have a closed-form in term of eta quotients but I haven't found it yet.
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