Let F be a real quadratic number field with discriminant D and \(\mathcal {O}_F\) the ring of integers in F. Let \(\chi _F\) be the Dirichlet character associated to \(F/\mathbb {Q}\) . Write \(L(\chi _F,s)\) for the Dirichlet L-function of \(\chi _F\) . By an induction argument for imprimitive Dirichlet L-values, we get several 2-divisibility results on \(L(\chi _F,-1)\) when D has arbitrarily finitely many prime divisors. As an application, by making use of the Birch–Tate formula for F, we determine the 2-primary part for the second K group \(K_2\mathcal {O}_F\) . We also give a new proof for an old theorem of Browkin and Schinzel.
