Let \(L(s, \chi _1), \ldots , L(s, \chi _N)\) be primitive Dirichlet L-functions different from the Riemann zeta function. Under suitable hypotheses we prove that any linear combination \(a_1\log |L(\rho ,\chi _1)|+\dots +a_N\log |L(\rho ,\chi _N)|\) has an approximately normal distribution as \(T\rightarrow \infty \) with mean 0 and variance \( \tfrac{1}{2} \big ({a_1}^2+\dots +{a_N}^2\big )\log \log T.\) Here \(a_1, a_2, \ldots , a_N \in \mathbb {R}\) , and \(\rho \) runs over the nontrivial zeros of the zeta function with \(0< \Im \rho \le T\) . From this we deduce that the vectors \(\big (\log |L(\rho ,\chi _1)|/\sqrt{ \frac{1}{2} \log \log T}, \ldots , \log |L(\rho ,\chi _N)|/\sqrt{\frac{1}{2} \log \log T}\,\big )\) have approximately an N-variate normal distribution whose components are approximately mutually independent as \(T\rightarrow \infty \) . We apply these results to study the proportion of the \(\rho \) that are zeros or a-values of linear combinations of the form \(c_1 L(\rho , \chi _1)+ \cdots + c_N L(\rho , \chi _N)\) with complex \(c_i\) as coefficients.
