  In a previous article we had proved an algebraicity result for the central
critical value for L-functions for GL(n) x GL(n-1) over Q assuming the validity
of a nonvanishing hypothesis involving archimedean integrals. The purpose of
this article is to generalize that result for all critical values for
L-functions for GL(n) x GL(n-1) over any number field F. Binyong Sun has
recently proved that nonvanishing hypothesis and so the results of this article
are unconditional. Using such results for the case of GL(3) x GL(2), new
unconditional algebraicity results for the special values of symmetric cube
L-functions for GL(2) over F have been proved.
