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Fungrim entry: 629f70

L ⁣(s,χq(1,))=ζ ⁣(s)pq(11ps)L\!\left(s, \chi_{q}(1, \cdot)\right) = \zeta\!\left(s\right) \prod_{p \mid q} \left(1 - \frac{1}{{p}^{s}}\right)
Assumptions:qZ1andsCq \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, s \in \mathbb{C}
TeX:
L\!\left(s, \chi_{q}(1, \cdot)\right) = \zeta\!\left(s\right) \prod_{p \mid q} \left(1 - \frac{1}{{p}^{s}}\right)

q \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, s \in \mathbb{C}
Definitions:
Fungrim symbol Notation Short description
DirichletLL ⁣(s,χ)L\!\left(s, \chi\right) Dirichlet L-function
DirichletCharacterχq(,)\chi_{q}(\ell, \cdot) Dirichlet character
RiemannZetaζ ⁣(s)\zeta\!\left(s\right) Riemann zeta function
PrimeProductpf ⁣(p)\prod_{p} f\!\left(p\right) Product over primes
Powab{a}^{b} Power
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
CCC\mathbb{C} Complex numbers
Source code for this entry:
Entry(ID("629f70"),
    Formula(Equal(DirichletL(s, DirichletCharacter(q, 1)), Mul(RiemannZeta(s), PrimeProduct(Parentheses(Sub(1, Div(1, Pow(p, s)))), p, Divides(p, q))))),
    Variables(q, s),
    Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(s, CC))))

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

2019-08-21 11:44:15.926409 UTC