  The s-wave pion-pion ($\pi\pi$) scattering lengths are computed below the
inelastic threshold by the L\"uscher technique with pion masses ranging from
240 MeV to 463 MeV. In the Asqtad-improved staggered fermion formulation, we
calculate the $\pi\pi$ four-point functions for the I=0 and 2 channels with
"moving" wall sources without gauge fixing, and analyze them at the
next-to-leading order in the continuum three-flavor chiral perturbation theory.
At the physical pion mass, we secure the s-wave $\pi\pi$ scattering lengths as
$m_\pi a_{\pi\pi}^{I=0} = 0.214(4)(7)$ and $m_\pi a_{\pi\pi}^{I=2} =
-0.04430(25)(40)$ for the I=0 and 2 channels, respectively, where the first
uncertainties are statistical and second ones are our estimates of several
systematic effects. Our lattice results for the s-wave $\pi\pi$ scattering
lengths are in well accordance with available experimental reports and
theoretical forecasts at low momentum. A basic ingredient in our study for the
I=0 case is properly incorporating disconnected diagram. These lattice
computations are carried out with the MILC 2+1 flavor gauge configurations at
two lattice spacings $a \approx 0.15$ and 0.12 fm.
