The relative Langlands duality posed by Ben-Zvi-Sakellaridis-Venkatesh predicts that periods of automorphic form can sometimes equal to sum of several special values of automorphic L-functions rather than a single one. In this talk, I will present explicit evidence for this phenomenon. I will discuss families of automorphic periods on general linear groups and classical groups that are designed to detect specific types of Eisenstein series. These periods indeed evaluate to finite sums of L-values. This is joint work with Guodong Xi.