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In mathematics, an L-function is a meromorphic function on the complex plane, associated to one out of several categories of mathematical objects. An L-series is a power series, usually convergent on a half-plane, that may give rise to an L-function via analytic continuation. The theory of ''L''-functions has become a very substantial, and still largely conjectural, part of contemporary analytic number theory. In it, broad generalisations of the Riemann zeta function and the ''L''-series for a Dirichlet character are constructed, and their general properties, in most cases still out of reach of proof, are set out in a systematic way. == Construction == We distinguish at the outset between the L-series, an infinite series representation (for example the Dirichlet series for the Riemann zeta function), and the ''L''-function, the function in the complex plane that is its analytic continuation. The general constructions start with an ''L''-series, defined first as a Dirichlet series, and then by an expansion as an Euler product indexed by prime numbers. Estimates are required to prove that this converges in some right half-plane of the complex numbers. Then one asks whether the function so defined can be analytically continued to the rest of the complex plane (perhaps with some poles). It is this (conjectural) meromorphic continuation to the complex plane which is called an ''L''-function. In the classical cases, already, one knows that useful information is contained in the values and behaviour of the ''L''-function at points where the series representation does not converge. The general term ''L''-function here includes many known types of zeta-functions. The Selberg class is an attempt to capture the core properties of ''L''-functions in a set of axioms, thus encouraging the study of the properties of the class rather than of individual functions. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「L-function」の詳細全文を読む スポンサード リンク
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