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Let      ψ  ( x ) = Σ  n = 1 ∞  e −  n 2 π  x \psi (x) = \Sigma _{n = 1}^\infty {e^{ - {n^2}\pi x}}      , and      χ  ( u ) = e u / 2 ( 1 + 2 ψ  ( e 2 u ) ) \chi (u) = {e^{u/2}}(1 + 2\psi ({e^{2u}}))      . The divergent integral      2 ∫  0 ∞  χ  ( u ) cos 2\smallint _0^\infty \chi (u)\cos        tu du  is shown to be summable for certain complex values of  t  to the function      2 ξ  ( s ) / s ( s −  1 ) = π  −  s / 2 Γ  ( 1 / 2 s ) ζ  ( s ) 2\xi (s)/s(s - 1) = {\pi ^{ - s/2}}\Gamma (1/2s)\zeta (s)      , where      s = 1 / 2 + i t s = 1/2 + it      , and      ζ  ( s ) \zeta (s)       is the zeta-function of Riemann. The values of a resulting approximation to      2 ξ  ( s ) / s ( s −  1 ) 2\xi (s)/s(s - 1)       are computed and its zeros located.

On an integral summable to 2𝜉(𝑠)/(𝑠(𝑠-1))