Let \(\Pi \) be a regular algebraic cuspidal automorphic representation (RACAR) of \(\textrm{GL}_3(\mathbb {A}_{\mathbb {Q}})\) . When \(\Pi \) is p-nearly-ordinary for the maximal standard parabolic with Levi \(\textrm{GL}_1 \times \textrm{GL}_2\) , we construct a p-adic L-function for \(\Pi \) . More precisely, we construct a (single) bounded measure \(L_p(\Pi )\) on \(\mathbb {Z}_p^{\times }\) attached to \(\Pi \) , and show it interpolates all the critical values \(L(\Pi \times \eta ,-j)\) at p in the left-half of the critical strip for \(\Pi \) (for varying \(\eta \) and j). This proves conjectures of Coates–Perrin-Riou and Panchishkin in this case. We also prove a corresponding result in the right half of the critical strip, assuming near-ordinarity for the other maximal standard parabolic. Our construction uses the theory of spherical varieties to build a “Betti Euler system”, a norm-compatible system of classes in the Betti cohomology of a locally symmetric space for \(\textrm{GL}_3\) . We work in arbitrary cohomological weight, allow arbitrary ramification at p along the Levi factor of the standard parabolic, and make no self-duality assumption. We thus give the first constructions of p-adic L-functions for RACARs of \(\textrm{GL}_n(\mathbb {A}_{\mathbb {Q}})\) of ‘general type’ (i.e. those that do not arise as functorial lifts) for any \(n >2\) .
