Fungrim entry: b788a1

\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:

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G^{\text{Primitive}}_{q} \;\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^{\text{Primitive}}_{q} \;\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
`DirichletLambda`	$\Lambda\!\left(s, \chi\right)$	Completed Dirichlet L-function
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)
`Gamma`	$\Gamma(z)$	Gamma function
`DirichletL`	$L\!\left(s, \chi\right)$	Dirichlet L-function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`PrimitiveDirichletCharacters`	$G^{\text{Primitive}}_{q}$	Primitive Dirichlet characters with given modulus
`CC`	$\mathbb{C}$	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))))))

Topics using this entry

Dirichlet L-functions