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We study the distribution of large (and small) values of several families of  L -functions on a line Re( s )= σ , where. We consider the Riemann zeta function  ζ ( s ) in the  t -aspect, Dirichlet  L -functions in the  q -aspect, and  L -functions attached to primitive holomorphic cusp forms of weight 2 in the level aspect. For each family, we show that the  L -values can be very well modeled by an adequate random Euler product, uniformly in a wide range. We also prove new  Ω -results for quadratic Dirichlet  L -functions (predicted to be best possible by the probabilistic model) conditionally on generalized Riemann hypothesis, and other results related to large moments of  ζ ( σ + it ).

On the Distribution of Extreme Values of Zeta and L -Functions in the Strip ½ < σ < 1