  It is known that for a dual pair of unitary groups with equal size, zeta
integrals arising from Rallis inner product formula give the central values of
certain automorphic L-functions, which have applications to arithmetic. In this
paper we explicitly calculate archimedean zeta integrals of this type for
U(2,1). In particular we compute the matrix coefficients of Weil
representations using joint harmonics in the Fock model, and those of discrete
series using Schmid operators.
