The construction of p-adic L-functions forms an important first step for Iwasawa-Greenberg main conjectures and variants, leading to insight on the structure of corresponding Selmer groups consistent with the predictions of Birch and Swinnerton-Dyer. I will explain such a starting point for developing higher-rank analogues of the existing theory for \(GL_2\) in anticyclotomic towers. To be more precise, fix \(n \geq 2\) an integer, and K a CM field. Let \(\pi\) be a cuspidal automorphic representation of \(GL_n(\mathbb A_K)\) which is ordinary at a given prime P of K. Using the theory of Eulerian integral presentations with essential Whittaker vectors, I will explain how to construct from some well-chosen pure tensor a p-adic interpolation series for algebraic parts of the degree n L-function \(L(s, \pi \otimes \chi)\) at critical points \(s= s_0\), where \(\chi\) varies over ring class characters factoring through the anticyclotomic tower. If time permits, then I will also outline one or two potential applications.