Hadjicostas's formula relates the $\Gamma$ and $\zeta$ functions with a special double integral:
$$ \int _{0}^{1}\int _{0}^{1}{\frac {1-x}{1-xy}}(-\log(xy))^{s}\,dx\,dy=\Gamma (s+2)\left(\zeta (s+2)-{\frac {1}{s+1}}\right), $$
for $\Re(s)>-2$ and $s\neq-1$. Plugging in $s=1$ yields the special value
$$ \int _{0}^{1}\int _{0}^{1}{\frac {1-x}{1-xy}}(-\log(xy))\,dx\,dy=2\zeta(3)-1, $$
which resembles the double integral used in Beukers' proof of the irrationality of $\zeta(3)$:
$$ \int _{{0}}^{{1}}\int _{{0}}^{{1}}{\frac {-\log(xy)}{1-xy}}{\tilde {P_{{n}}}}(x){\tilde {P_{{n}}}}(y)dxdy, $$
where ${\tilde {P_{{n}}}}(x)$ are shifted Legendre polynomials.
Is it possible to conclude the irrationality of $\zeta(3)$ solely based on $$ \int _{0}^{1}\int _{0}^{1}{\frac {1-x}{1-xy}}(-\log(xy))\,dx\,dy=2\zeta(3)-1? $$