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In [[mathematics]], the [[L-function]]s of [[number theory]] are expected to have certain '''[[functional equation]]s''', as one of their characteristic properties. There is an elaborate theory of what these should be; much of it still conjectural. For example, the [[Riemann zeta function]] has a functional equation relating its value at the [[complex number]] ''s'' with its value at 1 − ''s''. In every case this relates to some value ζ(''s'') that is only defined by [[analytic continuation]] from the [[infinite series]] definition. That is, writing as is conventional σ for the real part of ''s'', the functional equation relates the cases
:σ > 1 and σ < 0,
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