Congruences for odd class numbers of quadratic fields with odd discriminant

For any distinct two primes p(1) equivalent to p(2) equivalent to 3 (mod 4), let h(-p(1)), h(-p(2)) and h(p(1)p(2)) be the class numbers of the quadratic fields Q(root-p(1)), Q(root-p(2)) and Q(root p(1)p(2)), respectively. Let omega(p1p2) := (1 + root p(1)p(2))/2 and let Psi(omega(p1p2)) be the Hirzebruch sum of omega(p1p2). We show that h(-p(1))h(-p(2)) equivalent to h(p(1)p(2))Psi(omega(p1p2))/n (mod 8), where n = 6 (respectively, n = 2) if min p1, p2 > 3 (respectively, otherwise). We also consider the real quadratic order with conductor 2 in Q(root p(1)p(2)).