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The right-hand side of the well known equation:

$$\displaystyle \pi^\frac{-s}{2}\,\Gamma\left(\frac{s}{2}\right)\, \zeta(s) =\int_1^{\infty} \left({x}^{\frac{s}{2}} + {x}^{\frac{1-s}{2}}\right)\,\frac{\psi(x)}{x} \text{d}x - \frac{1}{s\,(1-s)}$$

with $\displaystyle \psi(x)=\sum_{n=1}^{\infty}e^{-\pi\,n^2\,x}$, can be decomposed as $A(s)+A(1-s)$ when:

$$A(s):=\int_1^{\infty} \frac{{x}^{\frac{s}{2}} \,\psi(x)}{x} \text{d}x - \frac{1}{s}$$

Note that $A(s)=\overline{A(\overline{s})}$ and the reflection formula is $A(s)=\pi^\frac{-s}{2}\,\Gamma\left(\frac{s}{2}\right)\, \zeta(s)-A(1-s)$. This implies that when $s=\rho$, it follows that $A(\rho)=-A(1-\rho)$.

$A(s)$ also has complex roots, that seem to increase quite regularly (truncated at 3 decimals):

$15.809\pm11.848\,i, 19.574\pm19.240\,i, 22.660\pm25.536\,i, 25.364\pm31.263\,i, 27.815\pm36.621\,i, 30.083\pm41.713\,i, 32.210\pm46.600\,i, 34.224\pm51.322\,i, 36.145\pm55.907\,i, 37.988\pm60.377\,i, 39.763\pm64.742\,i, 41.480\pm69.027\,i, 43.145\pm73.230\,i, 44.766+77.552\,i, 46.375\pm81.286\,i, 47.511\pm84.415\,i,\dots $

Note that since these zeros seem to grow increasingly further away from the critical strip, it follows from the reflection formula above ($\rho$'s exist only in the strip) that $A(s) \ne A(1-s)$ at these zeros.


Is anything known about these zeros? Have these been studied before?

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