Average Values of L-functions anywhere in the critical strip

We will talk about new phenomena observed when studying the first moment of cubic L-functions at any s in the critical strip. We show that there is a phase transition in the moment at s=1/3, and an interesting symmetry in the moment happens between s>1/3 and s<1/3. In particular, at s=1/2, we improve the known asymptotic of Luo and prove the existence of a new secondary term. We also investigate these phenomena in general for higher order L-functions, and look at it from a random matrix theory perspective.