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We obtain various new limit theorems for random walks on SL_2(C) under lowmoment conditions. For non-elementary measures with a finite second moment, weprove a Local Limit Theorem for the norm cocycle, yielding the optimal versionof a theorem of E. Le Page. For measures with a finite third moment, we obtainthe Local Limit Theorem for the matrix coefficients, improving a recent resultof Grama-Quint-Xiao and the authors, and Berry-Esseen bounds with optimal rate$O(1 / \sqrt n)$ for the norm cocycle and the matrix coefficients. The maintool is a detailed study of the spectral properties of the Markov operator andits purely imaginary perturbations acting on different function spaces. Weintroduce, in particular, a new function space derived from the Sobolev space$W^{1,2}$ that provides uniform estimates.

Random walks on SL_2(C): spectral gap and limit theorems