Dihedral quartic approximations and series for π

Imaginary quadratic fields with class groups that have C(4) as a subgroup are analyzed in depth, and the units of associated dihedral quartic fields are thereby evaluated using Epstein zeta functions. Emphasis is placed on extreme examples such as Q(√−3502) which is probably the last case having an even discriminant and C(4) × C(4) as its class group. These extreme examples lead to very remarkable approximations and series for π such as π= 6 3502 log (2π)+7.37×10 −82 where u is the product of four, rather simple, quartic units. The approximations and series relate to Baker's theory of linear forms in logarithms and to certain modular identities. Related topics are discussed briefly.