Congruences for odd class numbers of quadratic fields with odd discriminant

For any distinct two primes \(p_1\equiv p_2\equiv 3\) \((\text {mod }4)\), let \(h(-p_1)\), \(h(-p_2)\) and \(h(p_1p_2)\) be the class numbers of quadratic fields \(\mathbb {Q}(\sqrt{-p_1})\), {Q}(\sqrt{-p_2})\) {Q}(\sqrt{p_1p_2})\), respectively. Let \(\omega _{p_1p_2}:=(1+\sqrt{p_1p_2})/2\) \(\varPsi (\omega _{p_1p_2})\) Hirzebruch sum _{p_1p_2}\). We show that \(h(-p_1)h(-p_2)\equiv h(p_1p_2)\varPsi _{p_1p_2})/n\) }8)\), where \(n=6\) (respectively, \(n=2\)) if \(min {p1, p2} > otherwise). also consider real order with conductor 2 in {Q}(\sqrt{p_1p_2})\).