There are only seven fundamental d=4p with class number 6, namely p=29,53,61,109,157,277,397. To prevent clutter, only the first three will be stated. Given the Kronecker symbol (dm), we propose
K(k29)=√2π2√1161(x1)1/3(y21+√y41+42)3/4(116∏m=1[Γ(m116)](−116m))1/12K(k53)=√2π2√2121(x2)1/3(y22+√y42+42)3/4(212∏m=1[Γ(m212)](−212m))1/12K(k61)=√2π2√2441(x3)1/3(y23+√y43+366)3/4(244∏m=1[Γ(m244)](−244m))1/12
where the xn are the real roots of the following cubics x3−5x2−3x−1=0x3−15x2−x−1=0x3−27x2−5x−1=0
while the yn are also the real roots of cubics y3−y2−4y−4=0y3−7y2+13y−11=0y3−6y2−27y−54=0
and similarly for the other p.
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