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We study the value-distribution of Dirichlet L-functions L(s, χ) in the half-plane σ = s > 1/2. The main result is that a certain average of the logarithm of L(s, χ) with respect to χ, or of the Riemann zeta-function ζ(s) with respect to s, can be expressed as an integral involving a density function, which depends only on σ and can be explicitly constructed. Several mean-value estimates on L-functions are essentially used in the proof in the case 1/2 < σ ≤ 1.