  We prove an algebraicity result for the central critical value of certain
Rankin-Selberg L-functions for GL(n) x GL(n-1). This is a generalization and
refinement of some results of Harder, Kazhdan-Mazur-Schmidt, Mahnkopf, and
Kasten-Schmidt. As an application of this result, we prove algebraicity results
for certain critical values of the fifth and the seventh symmetric power
L-functions attached to a holomorphic cusp form. Assuming Langlands
functoriality one can prove similar algebraicity results for the special values
of any odd symmetric power L-function. We also prove a conjecture of Blasius
and Panchishkin on twisted L-values in some cases. We comment on the
compatibility of our results with Deligne's conjecture on the critical values
of motivic L-functions. These results, as in the above mentioned works, are, in
general, based on a nonvanishing hypothesis on certain archimedean integrals.
