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Since we have

$\Gamma(1/2+it)=\sqrt{\pi/\cosh(\pi t)}\exp[i(2 \vartheta(t)+t \log(2\pi)+\arctan(\tanh(\pi t/2)))]$

where $\vartheta(t)$ is the Riemann Siegel function. The zeros on the critical line have ordinates the zeros of

the cosine or sine of the real function

$2 \vartheta(t)+t \log(2\pi)+\arctan(\tanh(\pi t/2))$

But there are real zeros, for example there is one at $s = 4.0260426340124070065475\dots$

X-ray:

Since we have

$\Gamma(1/2+it)=\sqrt{\pi/\cosh(\pi t)}\exp[i(2 \vartheta(t)+t \log(2\pi)+\arctan(\tanh(\pi t/2)))]$

where $\vartheta(t)$ is the Riemann Siegel function. The zeros on the critical line have ordinates the zeros of

the cosine or sine of the real function

$2 \vartheta(t)+t \log(2\pi)+\arctan(\tanh(\pi t/2))$

But there are real zeros, for example there is one at $s = 4.0260426340124070065475\dots$

X-ray:

Since we have

$\Gamma(1/2+it)=\sqrt{\pi/\cosh(\pi t)}\exp[i(2 \vartheta(t)+t \log(2\pi)+\arctan(\tanh(\pi t/2)))]$

where $\vartheta(t)$ is the Riemann Siegel function. The zeros on the critical line have ordinates the zeros of

the cosine or sine of the real function

$2 \vartheta(t)+t \log(2\pi)+\arctan(\tanh(\pi t/2))$

But there are real zeros, for example there is one at $s = 4.0260426340124070065475\dots$

Since we have

$\Gamma(1/2+it)=\sqrt{\pi/\cosh(\pi t)}\exp[i(2 \vartheta(t)+t \log(2\pi)+\arctan(\tanh(\pi t/2)))]$

where $\vartheta(t)$ is the Riemann Siegel function. The zeros on the critical line have ordinates the zeros of

the cosine or sine of the real function

$2 \vartheta(t)+t \log(2\pi)+\arctan(\tanh(\pi t/2))$

But there are real zeros, for example there is one at $s = 4.0260426340124070065475\dots$

X-ray:

Since we have

$\Gamma(1/2+it)=\sqrt{\pi/\cosh(\pi t)}\exp[i(2 \vartheta(t)+t \log(2\pi)+\arctan(\tanh(\pi t/2)))]$

where $\vartheta(t)$ is the Riemann Siegel function.

we may deduce (with some effort) that there is no other complex zero, and those on the critical line have ordinates the zeros of

the cosine or sine of the real function

$2 \vartheta(t)+t \log(2\pi)+\arctan(\tanh(\pi t/2))$

But there are real zeros, for example there is one at $s = 4.0260426340124070065475\dots$

Since we have

$\Gamma(1/2+it)=\sqrt{\pi/\cosh(\pi t)}\exp[i(2 \vartheta(t)+t \log(2\pi)+\arctan(\tanh(\pi t/2)))]$

where $\vartheta(t)$ is the Riemann Siegel function. The zeros on the critical line have ordinates the zeros of

the cosine or sine of the real function

$2 \vartheta(t)+t \log(2\pi)+\arctan(\tanh(\pi t/2))$

But there are real zeros, for example there is one at $s = 4.0260426340124070065475\dots$