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Fungrim entry: 6c3fff

L ⁣(1,χ)={~,χ=χq(1,)lims1L ⁣(s,χ),otherwiseL\!\left(1, \chi\right) = \begin{cases} {\tilde \infty}, & \chi = \chi_{q}(1, \cdot)\\\lim_{s \to 1} L\!\left(s, \chi\right), & \text{otherwise}\\ \end{cases}
Assumptions:qZ1andχGqq \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, \chi \in G_{q}
TeX:
L\!\left(1, \chi\right) = \begin{cases} {\tilde \infty}, & \chi = \chi_{q}(1, \cdot)\\\lim_{s \to 1} L\!\left(s, \chi\right), & \text{otherwise}\\ \end{cases}

q \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, \chi \in G_{q}
Definitions:
Fungrim symbol Notation Short description
DirichletLL ⁣(s,χ)L\!\left(s, \chi\right) Dirichlet L-function
UnsignedInfinity~{\tilde \infty} Unsigned infinity
DirichletCharacterχq(,)\chi_{q}(\ell, \cdot) Dirichlet character
ComplexLimitlimzaf ⁣(z)\lim_{z \to a} f\!\left(z\right) Limiting value, complex variable
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
DirichletGroupGqG_{q} Dirichlet characters with given modulus
Source code for this entry:
Entry(ID("6c3fff"),
    Formula(Equal(DirichletL(1, chi), Cases(Tuple(UnsignedInfinity, Equal(chi, DirichletCharacter(q, 1))), Tuple(ComplexLimit(DirichletL(s, chi), s, 1), Otherwise)))),
    Variables(q, chi),
    Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(chi, DirichletGroup(q)))))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-08-17 11:32:46.829430 UTC