Formula:DLMF:25.15:E5

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${\displaystyle{\displaystyle{\displaystyle L\left(1-s,\chi\right)=\frac{k^{s-1}\Gamma\left(s\right)}{(2\pi)^{s}}\*{\left({\mathrm{e}^{-\pi\mathrm{i}s/2}}+\chi(-1){\mathrm{e}^{\pi\mathrm{i}s/2}}\right)}\*G(\chi)L\left(s,\overline{\chi}\right)}}}$

Substitution(s)

${\displaystyle{\displaystyle{\displaystyle{\displaystyle G(\chi)=\sum_{r=1}^{k}\chi(r){\mathrm{e}^{2\pi\mathrm{i}r/k}}}}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle\chi}}}$ is a primitive character (mod ${\displaystyle{\displaystyle{\displaystyle k}}}$) &
${\displaystyle{\displaystyle{\displaystyle\overline{\chi}}}}$ is the complex conjugate of ${\displaystyle{\displaystyle{\displaystyle\chi}}}$

Proof

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Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle L}}}$ : Dirichlet ${\displaystyle{\displaystyle{\displaystyle L}}}$-function : http://dlmf.nist.gov/25.15#E1
${\displaystyle{\displaystyle{\displaystyle\Gamma}}}$ : Euler's gamma function : http://dlmf.nist.gov/5.2#E1
${\displaystyle{\displaystyle{\displaystyle\pi}}}$ : ratio of a circle's circumference to its diameter : http://dlmf.nist.gov/5.19.E4
${\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}$ : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
${\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}$ : imaginary unit : http://dlmf.nist.gov/1.9.i
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum

Bibliography

Equation (5), Section 25.15 of DLMF.

URL links

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