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25 Zeta and Related FunctionsRiemann Zeta Function

§25.4 Reflection Formulas

For s0,1,

25.4.1 ζ(1-s)=2(2π)-scos(12πs)Γ(s)ζ(s),
25.4.2 ζ(s)=2(2π)s-1sin(12πs)Γ(1-s)ζ(1-s).

Equivalently,

25.4.3 ξ(s)=ξ(1-s),

where ξ(s) is Riemann’s ξ-function, defined by:

25.4.4 ξ(s)=12s(s-1)Γ(12s)π-s/2ζ(s).

For s0,1 and k=1,2,3,,

25.4.5 (-1)kζ(k)(1-s)=2(2π)sm=0kr=0m(km)(mr)((ck-m)cos(12πs)+(ck-m)sin(12πs))Γ(r)(s)ζ(m-r)(s),

where

25.4.6 c=-ln(2π)-12πi.