# Reflection Formulas

## Reflection Formulas

$\displaystyle \Riemannzeta@{1-s} = 2 (2\cpi)^{-s} \cos@{\tfrac{1}{2} \cpi s} \EulerGamma@{s} \Riemannzeta@{s}$

Constraint(s): $\displaystyle {\displaystyle s \neq 0,1}$

$\displaystyle {\displaystyle \RiemannZeta@{s} = 2 (2\cpi)^{s-1} \sin@{\tfrac{1}{2} \cpi s} \EulerGamma@{1-s} \RiemannZeta@{1-s} }$
$\displaystyle {\displaystyle \RiemannXi@{s} = \RiemannXi@{1-s} }$
$\displaystyle {\displaystyle \RiemannXi@{s} = \tfrac{1}{2} s (s-1) \EulerGamma@{\tfrac{1}{2} s} \cpi^{-s/2} \RiemannZeta@{s} }$
$\displaystyle {\displaystyle \opminus^k \RiemannZeta^{(k)}@{1-s} = \frac{2}{(2\cpi)^s} \sum_{m=0}^k \sum_{r=0}^m \binom{k}{m} \binom{m}{r} \left( \realpart{(c^{k-m})} \cos@{\tfrac{1}{2} \cpi s} + \imagpart{(c^{k-m})} \sin@{\tfrac{1}{2} \cpi s} \right) \EulerGamma^{(r)}@{s} \RiemannZeta^{(m-r)}@{s} }$

Substitution(s): $\displaystyle {\displaystyle {\displaystyle c = -\ln@{2\cpi} - \tfrac{1}{2} \cpi \iunit }}$

Constraint(s): $\displaystyle {\displaystyle s \neq 0,1}$ &
$\displaystyle {\displaystyle k = 1,2,3,\dots}$