DLMF:25.11.E36 (Q7710)

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DLMF:25.11.E36
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    Statements

    Defining formula
    L ⁡ ( s , χ ) = ∑ n = 1 ∞ χ ⁢ ( n ) n s = k - s ⁢ ∑ r = 1 k - 1 χ ⁢ ( r ) ⁢ ζ ⁡ ( s , r k ) , Dirichlet-L 𝑠 𝜒 superscript subscript 𝑛 1 𝜒 𝑛 superscript 𝑛 𝑠 superscript 𝑘 𝑠 superscript subscript 𝑟 1 𝑘 1 𝜒 𝑟 Hurwitz-zeta 𝑠 𝑟 𝑘 {\displaystyle{\displaystyle L\left(s,\chi\right)=\sum_{n=1}^{\infty}\frac{% \chi(n)}{n^{s}}=k^{-s}\sum_{r=1}^{k-1}\chi(r)\zeta\left(s,\frac{r}{k}\right),}}
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    DLMF id
    DLMF:25.11.E36
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    constraint
    ℜ ⁡ s > 1 𝑠 1 {\displaystyle{\displaystyle\Re s>1}}
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    Symbols used
    Dirichlet $$L$$ -function
    Defining formula
    L ⁡ ( s , χ ) Dirichlet-L 𝑠 𝜒 {\displaystyle{\displaystyle L\left(\NVar{s},\NVar{\chi}\right)}}
    xml-id
    C25.S15.E1.m2adec
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    Hurwitz zeta function
    Defining formula
    ζ ⁡ ( s , a ) Hurwitz-zeta 𝑠 𝑎 {\displaystyle{\displaystyle\zeta\left(\NVar{s},\NVar{a}\right)}}
    xml-id
    C25.S11.E1.m2aajdec
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    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1asdec
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    Q12150
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C25.S1.XMD1.m1fdec
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    Q12148
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C25.S1.XMD3.m1sdec
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    Q12149
    Defining formula
    s 𝑠 {\displaystyle{\displaystyle s}}
    xml-id
    C25.S1.XMD7.m1ydec
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