Special values of L-functions by a Siegel-Weil-Kudla-Rallis formula

We study the arithmeticity of special values of L-functions attached to cuspforms which are Hecke eigenfunctions on hermitian quaternion groups Sp^∗(m,0) which form a reductive dual pair with G=O^∗(4n). For f₁ and f₂ two cuspforms on H, consider their theta liftings θf₁ and θf₂ on G. Then we compute a Rankin-Selberg type integral and obtain an integral representation of the standard L-function:

G〈θf₁⋅Es,θf₂〉=H〈f₁,f₂〉⋅Lstd(f₁,s).