  In a previous paper (arxiv:1409.7353), we introduced a regularized theta lift
for reductive dual pairs of the form $(Sp_4,O(V))$ with $V$ a quadratic vector
space over a totally real number field $F$. The lift takes values in the space
of $(1,1)$-currents on the Shimura variety attached to $GSpin(V)$, and we
proved that its values are cohomologous to currents given by integration on
special divisors against automorphic Green functions. In this paper, we will
evaluate the regularized theta lift on differential forms obtained as usual
(non-regularized) theta lifts. Using the Siegel-Weil formula and ideas of
Piatetskii-Shapiro and Rallis, we show that the result involves near central
special values of standard $L$-functions for $Sp_{4,F}$. An example concerning
products of Shimura curves will be given at the end of the paper.
