For an arbitrary number field \(K\) with ring of integers denoted by \({\mathcal O}\) we study \(\text{GL} (2,{\mathcal O})\)- and \(\text{SL} (2,{\mathcal O})\)- equivalence classes of binary quadratic forms \(\Phi\) defined over \({\mathcal O}\). After fixing the discriminant \(\Delta(\Phi)= \Delta\) for some \(\Delta\in{\mathcal O}\), we define two kinds of zeta functions, \[ \begin{aligned} P_ \Delta(s) &= \sum _{\substack{ [\Phi]\\ \Phi \text{ primitive}\\ \Delta(\Phi)= \Delta}}\;\sum _{\substack{ (x,y)\in ({\mathcal O}\times {\mathcal O})/ E(\Phi)\\ \Phi(x,y)\neq 0}} | N_{K/\mathbb{Q}} (\Phi (x,y))|^{-s} \quad (\Delta\neq \text{square}), \qquad \text{and}\\ \zeta_ \Delta (s) &= \sum _{\substack{ [\Phi]\\ \Delta(\Phi)= \Delta }}\;\sum _{\substack{ (x,y)\in ({\mathcal O}\times {\mathcal O})/ E(\Phi)\\ (x,y)_{\mathcal O} ={\mathcal O}\\ \Phi(x,y) \neq 0 }} | N_{K/\mathbb{Q}} (\Phi(x,y))|^{-s},\end{aligned} \] where the outer sum runs over all \(\text{GL}_ 2\)-equivalence classes of binary quadratic forms over \({\mathcal O}\) of discriminant \(\Delta\), and where \(E(\Phi)\) is the automorphism group of \(\Phi\). The first function \(P_ \Delta (s)\) is identified with the L-series attached to similarity classes of free proper modules of the unique free order of discriminant \(\Delta\), whereas \(\zeta_ \Delta(s)\) is described in terms of numbers of solutions to \(x^ 2\equiv \Delta\bmod 4n\) as \(n\) varies over \({\mathcal O}\), which are explicitly determined.