$m_l=0$ in Hydrogen atom of Scrodinger equation $m_l=0$ corresponds to s-orbital and the values associated are:
$$\lambda=2\pi r/m_l\to\infty$$
It means a wave of of infinite wavelength has a constant height at all values of $\phi$(azimuthal angle).
$$J_z=\pm hr/\lambda\to0\text{ also }J_z=\pm pr$$
which implies zero linear and angular momentum as $r\ne0$.
Now what does zero angular and linear momentum signify/mean?(I think the electron is at rest but it cannot be) 
 A: The plane wave – the wave function for a particle moving in a clear direction and with a sharp momentum, $\vec p$, is
$$ \psi (\vec x) = C \exp (i\vec p\cdot \vec x / \hbar ) $$
where $\exp(ia)=\cos a + i \sin a$. However, the dependence of an $s$-wave wave function on the angular direction is trivial (constant), 
$$\psi_s (r,\theta,\phi) = \psi(r). $$ 
There is no dependence. You may imagine this wave function to be a superposition of some waves moving in all directions of the space, and backwards.
Because the wave function doesn't depend on the angles, in particular, it doesn't depend on $\phi$, you may imagine that it is a wave with an infinitely long wavelength in the angular directions. Note that $\cos 0\phi = 1$. The shorter the wavelength, the higher momentum, and vice versa – and it sort of holds for the angular momentum and the wavelength measured in spherical or azimuthal coordinates and in radians, too.
The zero angular momentum means that there is no rotation. If such a (slowly moving) particle (atom) is absorbed by a gyroscope (near the axis), the rate of the gyroscope rotation won't change.
