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Let f be a holomorphic Hecke eigenform or a Hecke-Maass cusp form for the full modular group SL(2, Z). In this paper we shall use circle method to prove the Weyl exponent for GL(2) L-functions. We shall prove thatfor any ǫ > 0.Lindelöf hypothesis asserts that the exponent 1/4 + ǫ can be replaced by ǫ. Sub-convexity bound for ζ(s) was first proved by G. H. Hardy and J. E. Littlewood, and H. Weyl independently.

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The moments of quadratic Dirichlet L-functions over function fields have recently attracted much attention with the work of Andrade and Keating. In this article, we establish lower bounds for the mean values of the product of quadratic Dirichlet Lfunctions associated with hyperelliptic curves of genus g over a fixed finite field F q in the large genus limit. By using the idea of A. Florea [14], we also obtain their upper bounds. As a consequence, we find upper bounds of its derivatives. These lower and upper bounds give the correlation of quadratic Dirichlet L-functions associated with hyperelliptic curves with different transitions.