Fix a number field k with its adele ring A. Let G=O(n+3) be an orthogonal group of k-rank 1 and H=O(n+2) a k-anisotropic subgroup. We have previously described how to factor the global period (E_φ,F)_H = ∫_Hk\HAE_φ⋅\bar{F} of a spherical Eisenstein series E_φ of G against a cuspform F of H into an Euler product. Here, we describe how to evaluate the factors at even primes. When the local field is unramified, we carry out the computation in all cases. We show also concrete examples of the complete period when k = Q. The results are consistent with the Gross-Prasad conjecture.