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We recall the notion of a tower of Rankin-Selberg integrals, and two known towers, making observations of how the integrals within a tower may be related to one another via formal manipulations, and offering a heuristic for how the L-functions should be related to one another when the integrals are related in this way. We then describe three new integrals in a tower on the group E 6 , and find out which L-functions they represent. The heuristics also predict the existence of a fourth integral.doi:10.1090/s1079-6762-06-00160-0 fatcat:ejzrdlyv6zfqzjv6mqccxuk524