We study the physics of multiple M5-branes compactified on a hyperbolic 3-manifold. On the one hand, it leads to the 3d-3d correspondence which maps an 3d superconformal field theory to a pure Chern-Simons theory on the 3-manifold. On the other hand, it leads to a warped AdS4 geometry in M-theory holographically dual to the superconformal field theory. Combining the holographic duality and the 3d-3d correspondence, we propose a conjecture for the large N limit of the perturbative free energy of a Chern-Simons theory on hyperbolic 3-manifold. The conjecture claims that the tree, one-loop and two-loop terms all share the same N3 scaling behavior and are proportional to the volume of the 3-manifold, while the three-loop and higher terms are suppressed at large N. We prove the tree and one-loop parts of the conjecture. For the two and three-loop part, we test the conjecture numerically in a number of examples using Dimofte's state-integral model and find precise agreement.