  For an even positive integer $n$, we study rank-one Eisenstein cohomology of
the split orthogonal group ${\rm O}(2n+2)$ over a totally real number field
$F.$ This is used to prove a rationality result for the ratios of successive
critical values of degree-$2n$ Langlands $L$-functions associated to the group
${\rm GL}_1 \times {\rm O}(2n)$ over $F$. The case $n=2$ specializes to
classical results of Shimura on the special values of Rankin - Selberg
$L$-functions attached to a pair of Hilbert modular forms.
