Let \(f(z)=\sum _{n=0}^{\infty }a_{n}{\mathbf {e}}(nz),g(z)=\sum _{n=0}^{\infty }b_{n}{\mathbf {e}}(nz)\ ({\mathbf {e}}(z)=e^{2\pi \sqrt{-1}z})\) be holomorphic modular forms for \(\Gamma _{0}(N)\) of integral weight or half integral weight, where their weights or characters are not necessarily equal to each other. We show that \(L(s;f,g):=\sum _{n=1}^{\infty }a_{n}{\overline{b}}_{n}n^{-s}\) extends meromorphically to the whole s plane, and that it satisfies some functional equation. Some residues of the L-function or some special values are expressed in terms of the Petersson scalar product. Applications to quadratic forms are included.

中文翻译：

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Petersson标量产品和L函数源自模块化形式

令\（f（z）= \ sum _ {n = 0} ^ {\ infty} a_ {n} {\ mathbf {e}}（nz），g（z）= \ sum _ {n = 0} ^ {\ infty} b_ {n} {\ mathbf {e}}（nz）\（{\ mathbf {e}}（z）= e ^ {2 \ pi \ sqrt {-1} z}）\）是全纯的整数权重或整数一半的\（\ Gamma _ {0}（N）\）的模块化形式，其中它们的权重或字符不一定彼此相等。我们证明\（L（s; f，g）：= \ sum _ {n = 1} ^ {\ infty} a_ {n} {\ overline {b}} _ {n} n ^ {-s} \ ）亚纯地延伸到整个s平面，并且满足某些函数方程。L功能的某些残基或某些特殊值以彼得森标量积表示。包括二次形式的应用。