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Let L(s, Φ) be an automorphic L-function for a Bianchi group defined via an imaginary quadratic field with discriminant d < 0. We give an upper bound for the absolute value of L(s, Φ) in terms of J(s), the Laplace eigenvalue and the discriminant d. The bound as d → ∞ presents a new aspect in the study of L-functions.

Estimates of automorphic $L$-functions in the discriminant-aspect