p-adic L-functions for GL(2n) in Shalika families with maximal variation

I will describe the construction of a (n+2)-variable p-adic L-function for p-adic families of automorphic representations of GL(2n) which admit a Shalika model, which is the maximal amount of p-adic variation that one expects in this setting. For n=2, this is one of the ingredients used in the work of Loeffler--Zerbes on the Bloch--Kato conjecture for automorphic representations of GSp(4). This is joint work with D. Barrera, M. Dimitrov, A. Jorza and C. Williams.