  The main objects of study in this article are two classes of Rankin-Selberg
L-unctions, namely L(s, f \times g) and L(s, sym^2(g) \times sym^2(g)), where
f, g are newforms, holomorphic or of Maass type, on the upper half plane, and
sym^2(g) denotes the symmetric square lift of g to GL(3). We prove that in
general, i.e., when these L-functions are not divisible by L-functions of
quadratic characters (such divisibility happening rarely), they do not admit
any Landau-Siegel zeros. These zeros, which are real and close to s=1, are
highly mysterious and are not expected to occur. There are corollaries of our
result, one of them being a strong lower bound for special value at s=1, which
is of interest both geometrically and analytically. One also gets this way a
good bound on the norm of sym^2(g).
