Given an imaginary quadratic field $\mathbb{Q}(\sqrt{-D})$, where $D$ is a Heegner number (1, 2, 3, 7, 11, 19, 43, 67, 163), what is the probability that two randomly selected elements of that fields' ring of integers are coprime? mathworld: For Gaussian integers it's $\frac{6}{\pi^{2}K}$, where $K$ is Catalan's constant, and for Eisenstein integers it's $\frac{6\cdot9}{\pi^{2}[\psi_{1}(1/3)-\psi_{1}(2/3)]}$, where $\psi_{1}$ is the trigamma function.
It might be answered by the Porubský, S. "On the Probability That K Generalized Integers Are Relatively H-Prime." Colloq. Math. 45, 91-99, 1981. reference on the mathworld page, but that's not media I have access to easily.
Remember that in the case of $\mathbb Z$, the answer is $6/\pi^2$, which arises conceptually as $1/\zeta(2)$.
So won't the same argument for $K$ (of class number one) give the answer $1/\zeta_K(2)$, where $\zeta_K$ is the Dedekind $\zeta$-function of $K$?
We have the factorization $\zeta_K(2) = \zeta(2) L(2,\chi) = \dfrac{pi^2}{6}L(2,\chi)$ (where $\chi$ is the quad. char. attached to $K$), and so the probability should be $\dfrac{6}{\pi^2} \dfrac{1}{L(2,\chi)}.$
(The formulas you state are special cases of this one.)