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I believe that I can to use the equation (1) (from reference [1]), a well known equation involving the Gamma function, Riemann zeta function and the theta function $\psi(x)=\sum_{n=1}^\infty e^{-n^2\pi x}$, for relating $\Gamma(\rho_k)$ and $\zeta(3+2\omega_k)$, where $\rho_k=\frac{1}{2}+i\omega_k$ is a (fixed) nontrivial zero of Riemann zeta function, and is taken $3+2\omega_k=s_k$, as this manner $$\Gamma(\rho_k)\pi^{\frac{-s_k}{2}}\zeta(3+2\omega_k)=\int_{0}^\infty\psi(x)x^{\frac{s_k}{2}}\frac{dx}{x}.$$

Question. Can you compute (I believe that in terms of the given real $\omega_k$) the values $\Gamma(\rho_k)$ or $\zeta(3+2\omega_k)$, when we assume that $\rho_k=\frac{1}{2}+i\omega_k$ is a nontrivial (fixed) zero of Riemann zeta function? Thanks in advance.

I don't know if using the functionals equations my question could be easily answered, or is a difficult question, and I excuse to compute $\Gamma(\rho_k)$ or $\zeta(3+2\omega_k)$, for the case it is difficult compute both values.

References:

[1] Edwards, Riemann Zeta function, equation (1) in page 15, I provide to you (I hope that you can read it) a free book.google.es link https://books.google.es/books?id=ruVmGFPwNhQC&pg=PA15&hl=es&source=gbs_toc_r&cad=3#v=onepage&q&f=false

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  • $\begingroup$ I am waiting useful answers for this Mathematics Stack Exchange, I don't know if using functional equations is easily answered my question; I don't know is one of the terms in the equation that I've written is easy to compute. My only goal with this question is encourage to me in the study of mathematics, thanks all users. $\endgroup$ – user243301 Dec 5 '15 at 12:50
  • $\begingroup$ I wish you Math Christmas and a Happy New $\frac{\left(\zeta(2)^2-\zeta(0)\zeta(4)\right)^3}{\zeta(0)^6\zeta(4)\zeta(8)}.$ After the bounty question was answered I will delete this last comment. Thanks $\endgroup$ – user243301 Dec 24 '15 at 16:16

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