Dedekind zeta function of a real quadratic field

Question

When looking at an imaginary quadratic field, its zeta function is not too difficult to explicitly define (ex: $ \zeta_{\mathbb{Q}(i)}(s) = \frac{1}{4} \sum_{(m,n) \in \mathbb{Z}^2\(0,0)} \frac{1}{(m^2+n^2)^s}$). Unfortunately, I realized that in any other field it's a lot harder to define the zeta function due to the fact that there's an infinitude of elements with equal norm due to the group of units. I attempted to find a closed form of $\zeta_{\mathbb{Q} (\sqrt 2)}(s)$ but couldn't find a way to create an index that omits elements having $(1+\sqrt2)^{\mathbb{Z}}$ (units) as a factor. Is there any way to find an expression for the zeta function of real quadratic fields similar to that of imaginary quadratic fields?