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The zeros and poles of standard automorphic $L$-functions attached torepresentations of classical groups are linked to the nonvanishing of lifts inthe theory of the theta correspondence. The results of this paper show thatwhen a cuspidal representation $\sigma$ of a symplectic group $G$ lifts to acuspidal representation $\pi = \theta_{\psi}(\sigma)$ of an orthogonal group$H$ attached to a quadratic space $V$ of dimension $m$, the Fouriercoefficients of automorphic forms in $\sigma$ are linked to periods ofautomorphic forms in $\pi$. Consequently, when our results are combined withthe Rallis inner product formula in the convergent range or the second termrange, we prove a special value formula for the standard automorphic$L$-function $L(s,\sigma,\chi_V)$ attached to $\sigma$ (and twisted by thecharacter $\chi_V$ of $V$) at the point $s_{m,2n}=m/2-(2n+1)/2$ in terms of theFourier coefficients of automorphic forms in $\sigma$ and periods of forms in$\pi$.

The Theta Correspondence, Periods of Automorphic Forms and Special Values of Standard Automorphic L-Functions