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Fungrim entry: b788a1

Λ ⁣(s,χ)=β(qπ)(s+a)/2Γ ⁣(s+a2)L ⁣(s,χ)   where a=1χ(1)2,  β={s(s1),q=11,otherwise\Lambda\!\left(s, \chi\right) = \beta {\left(\frac{q}{\pi}\right)}^{\left( s + a \right) / 2} \Gamma\!\left(\frac{s + a}{2}\right) L\!\left(s, \chi\right)\; \text{ where } a = \frac{1 - \chi(-1)}{2},\;\beta = \begin{cases} s \left(s - 1\right), & q = 1\\1, & \text{otherwise}\\ \end{cases}
Assumptions:qZ1  and  χGqprimitive  and  sC  and  s{{2n:nZ0},χ(1)=1{2n1:nZ0},χ(1)=1  and  not(q=1  and  s=1)q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G_{q}^{\text{primitive}} \;\mathbin{\operatorname{and}}\; s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; s \notin \begin{cases} \left\{ -2 n : n \in \mathbb{Z}_{\ge 0} \right\}, & \chi(-1) = 1\\\left\{ -2 n - 1 : n \in \mathbb{Z}_{\ge 0} \right\}, & \chi(-1) = -1\\ \end{cases} \;\mathbin{\operatorname{and}}\; \operatorname{not} \left(q = 1 \;\mathbin{\operatorname{and}}\; s = 1\right)
TeX:
\Lambda\!\left(s, \chi\right) = \beta {\left(\frac{q}{\pi}\right)}^{\left( s + a \right) / 2} \Gamma\!\left(\frac{s + a}{2}\right) L\!\left(s, \chi\right)\; \text{ where } a = \frac{1 - \chi(-1)}{2},\;\beta = \begin{cases} s \left(s - 1\right), & q = 1\\1, & \text{otherwise}\\ \end{cases}

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G_{q}^{\text{primitive}} \;\mathbin{\operatorname{and}}\; s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; s \notin \begin{cases} \left\{ -2 n : n \in \mathbb{Z}_{\ge 0} \right\}, & \chi(-1) = 1\\\left\{ -2 n - 1 : n \in \mathbb{Z}_{\ge 0} \right\}, & \chi(-1) = -1\\ \end{cases} \;\mathbin{\operatorname{and}}\;  \operatorname{not} \left(q = 1 \;\mathbin{\operatorname{and}}\; s = 1\right)
Definitions:
Fungrim symbol Notation Short description
DirichletLambdaΛ ⁣(s,χ)\Lambda\!\left(s, \chi\right) Completed Dirichlet L-function
Powab{a}^{b} Power
Piπ\pi The constant pi (3.14...)
GammaΓ(z)\Gamma(z) Gamma function
DirichletLL ⁣(s,χ)L\!\left(s, \chi\right) Dirichlet L-function
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
PrimitiveDirichletCharactersGqprimitiveG_{q}^{\text{primitive}} Primitive Dirichlet characters with given modulus
CCC\mathbb{C} Complex numbers
Source code for this entry:
Entry(ID("b788a1"),
    Formula(Equal(DirichletLambda(s, chi), Where(Mul(Mul(Mul(beta, Pow(Div(q, Pi), Div(Add(s, a), 2))), Gamma(Div(Add(s, a), 2))), DirichletL(s, chi)), Equal(a, Div(Sub(1, chi(-1)), 2)), Equal(beta, Cases(Tuple(Mul(s, Sub(s, 1)), Equal(q, 1)), Tuple(1, Otherwise)))))),
    Variables(q, chi, s),
    Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(chi, PrimitiveDirichletCharacters(q)), Element(s, CC), NotElement(s, Cases(Tuple(Set(Neg(Mul(2, n)), For(n), Element(n, ZZGreaterEqual(0))), Equal(chi(-1), 1)), Tuple(Set(Sub(Neg(Mul(2, n)), 1), For(n), Element(n, ZZGreaterEqual(0))), Equal(chi(-1), -1)))), Not(And(Equal(q, 1), Equal(s, 1))))))

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2020-04-08 16:14:44.404316 UTC