CBR photon density at equilibrium temperature

[Orion1]

CBR photon equilibrium temperature:
T_{\gamma} = 2.725 \; \text{K}

CBR photon density at equilibrium temperature T_{\gamma}:
n_{\gamma} = \frac{1}{\pi^2} {\left(\frac{k_B T_{\gamma}}{\hbar c}\right)}^3 \int_0^\infty \frac{x^2}{e^x - 1} dx

This is my solution for the integration equation on reference 2.

Integration:
\int_0^\infty \frac{x^2}{e^x - 1} dx = 2 \sum_{n=1}^\infty n^{-3} = 2 \zeta(3)

Complex variable:
\boxed{s = 3}

\boxed{\int_0^\infty \frac{x^2}{e^x - 1} dx = 2 \zeta(3)}

Where \zeta(s) is the Riemann zeta function.

Riemann zeta function:
\zeta(s) = \sum_{n=1}^\infty n^{-s}

The CBR photon density at equilibrium temperature T_{\gamma}:
\boxed{n_{\gamma} = \frac{2 \zeta(3)}{\pi^2} {\left(\frac{k_B T_{\gamma}}{\hbar c}\right)}^3}

\boxed{n_{\gamma} = 4.105 \cdot 10^{8} \; \frac{\text{photons}}{\text{m}^3}}

Are these equations correct?
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Reference:
http://en.wikipedia.org/wiki/Riemann_zeta_function"
http://en.wikipedia.org/wiki/Baryogenesis#Baryon_asymmetry_parameter"
http://en.wikipedia.org/wiki/Cosmic_microwave_background_radiation"

 

[Orion1]

CBR photon density at equilibrium temperature T_{\gamma}:
n_{\gamma} = \frac{1}{\pi^2} {\left(\frac{k_B T_{\gamma}}{\hbar c}\right)}^3 \int_0^\infty \frac{x^2}{e^x - 1} dx

How was this equation derived at this point?
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Reference:
http://en.wikipedia.org/wiki/Baryogenesis#Baryon_asymmetry_parameter"