Approximation of the Gamma function for small value It is well known that $\Gamma(u) \stackrel{u \to 0}{\sim} \frac{1}{u}$.
I am looking for more precise information on the behavior of $\Gamma(x)$ when $x$ is small, ie: $x\to 0$.
My question is then, are there accurate (for small value) inequalities for the Gamma function ? 
Any other information on the behavior is also very welcome.

$\Gamma(z)$ has a pole at zero, with Laurent series
$$
\frac{1}{z}-\gamma+ \left( {\frac {{\pi }^{2}}{12}}+{\frac {{\gamma}^{2}
}{2}} \right) z+ \left( -{\frac {\zeta  \left( 3 \right) }{3}}-{\frac 
{{\pi }^{2}\gamma}{12}}-{\frac {{\gamma}^{3}}{6}} \right) {z}^{2}+
 \left( {\frac {{\pi }^{4}}{160}}+{\frac {\zeta  \left( 3 \right) 
\gamma}{3}}+{\frac {{\pi }^{2}{\gamma}^{2}}{24}}+{\frac {{\gamma}^{4}
}{24}} \right) {z}^{3}+ \left( -{\frac {\zeta  \left( 5 \right) }{5}}-
{\frac {{\pi }^{4}\gamma}{160}}-{\frac {\zeta  \left( 3 \right) {\pi }
^{2}}{36}}-{\frac {\zeta  \left( 3 \right) {\gamma}^{2}}{6}}-{\frac {{
\pi }^{2}{\gamma}^{3}}{72}}-{\frac {{\gamma}^{5}}{120}} \right) {z}^{4
}+ \left( {\frac {61\,{\pi }^{6}}{120960}}+{\frac {\zeta  \left( 5
 \right) \gamma}{5}}+{\frac {{\pi }^{4}{\gamma}^{2}}{320}}+{\frac {
 \left( \zeta  \left( 3 \right)  \right) ^{2}}{18}}+{\frac {\zeta 
 \left( 3 \right) {\pi }^{2}\gamma}{36}}+{\frac {\zeta  \left( 3
 \right) {\gamma}^{3}}{18}}+{\frac {{\pi }^{2}{\gamma}^{4}}{288}}+{
\frac {{\gamma}^{6}}{720}} \right) {z}^{5}+O \left( {z}^{6} \right) 
$$
Of course $\gamma$ is Euler's constant and $\zeta$ is Riemann's zeta function.  
I wonder if it looks better if those $\pi^2$ and $\pi^4$ terms are written in terms of $\zeta(2)$ and $\zeta(4)$ and so on?  
$$
\frac{1}{z}-\gamma+ \left( {\frac {\zeta \left( 2 \right) }{2}}+{\frac {
{\gamma}^{2}}{2}} \right) z+ \left( -{\frac {\zeta  \left( 3 \right) 
}{3}}-{\frac {\zeta \left( 2 \right) \gamma}{2}}-{\frac {{\gamma}^{3}
}{6}} \right) {z}^{2}+ \left( {\frac {9\,\zeta \left( 4 \right) }{16}}
+{\frac {\zeta  \left( 3 \right) \gamma}{3}}+{\frac {\zeta \left( 2
 \right) {\gamma}^{2}}{4}}+{\frac {{\gamma}^{4}}{24}} \right) {z}^{3}+
 \left( -{\frac {\zeta  \left( 5 \right) }{5}}-{\frac {9\,\zeta
 \left( 4 \right) \gamma}{16}}-{\frac {\zeta  \left( 3 \right) \zeta
 \left( 2 \right) }{6}}-{\frac {\zeta  \left( 3 \right) {\gamma}^{2}}{
6}}-{\frac {\zeta \left( 2 \right) {\gamma}^{3}}{12}}-{\frac {{\gamma}
^{5}}{120}} \right) {z}^{4}+ \left( {\frac {61\,\zeta \left( 6
 \right) }{128}}+{\frac {\zeta  \left( 5 \right) \gamma}{5}}+{\frac {9
\,\zeta \left( 4 \right) {\gamma}^{2}}{32}}+{\frac { \left( \zeta 
 \left( 3 \right)  \right) ^{2}}{18}}+{\frac {\zeta  \left( 3 \right) 
\zeta \left( 2 \right) \gamma}{6}}+{\frac {\zeta  \left( 3 \right) {
\gamma}^{3}}{18}}+{\frac {\zeta \left( 2 \right) {\gamma}^{4}}{48}}+{
\frac {{\gamma}^{6}}{720}} \right) {z}^{5}+O \left( {z}^{6} \right) 
$$  
Maybe part of the $\zeta(4)$ should be a $\zeta(2)^2$ and similarly for the $\zeta(6)$ to make the pattern recognizable?
