Abstract Let Π be a regular algebraic cuspidal automorphic representation (RACAR) of GL₃ (A ₐ) GL 3 (A Q). When Π is p -nearly-ordinary for the maximal standard parabolic with Levi GL₁ GL₂ GL 1 × GL 2, we construct a p -adic L -function for Π. More precisely, we construct a (single) bounded measure Lₚ () L p (Π) on Zₚ^ Z p × attached to Π, and show it interpolates all the critical values L (, -j) L (Π × η, - j) at p in the left-half of the critical strip for Π (for varying η 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 GL₃ 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 GLₙ (A ₐ) GL n (A Q)
Loeffler et al. (Wed,) studied this question.