  We obtain various new limit theorems for random walks on SL_2(C) under low
moment conditions. For non-elementary measures with a finite second moment, we
prove a Local Limit Theorem for the norm cocycle, yielding the optimal version
of a theorem of E. Le Page. For measures with a finite third moment, we obtain
the Local Limit Theorem for the matrix coefficients, improving a recent result
of 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 main
tool is a detailed study of the spectral properties of the Markov operator and
its purely imaginary perturbations acting on different function spaces. We
introduce, in particular, a new function space derived from the Sobolev space
$W^{1,2}$ that provides uniform estimates.
