  We build on the formulation developed in Sridhar & Singh (JFM, 664, 265,
2010), and present a theory of the \emph{shear dynamo problem} for small
magnetic and fluid Reynolds numbers, but for arbitrary values of the shear
parameter. Specializing to the case of a mean magnetic field that is slowly
varying in time, explicit expressions for the transport coefficients,
$\alpha_{il}$ and $\eta_{iml}$, are derived. We prove that, when the velocity
field is non helical, the transport coefficient $\alpha_{il}$ vanishes. We then
consider forced, stochastic dynamics for the incompressible velocity field at
low Reynolds number. An exact, explicit solution for the velocity field is
derived, and the velocity spectrum tensor is calculated in terms of the
Galilean--invariant forcing statistics. We consider forcing statistics that is
non helical, isotropic and delta-correlated-in-time, and specialize to the case
when the mean-field is a function only of the spatial coordinate $X_3$ and time
$\tau\,$; this reduction is necessary for comparison with the numerical
experiments of Brandenburg, R{\"a}dler, Rheinhardt & K\"apyl\"a (ApJ, 676, 740,
2008). Explicit expressions are derived for all four components of the magnetic
diffusivity tensor, $\eta_{ij}(\tau)\,$. These are used to prove that the
shear-current effect cannot be responsible for dynamo action at small $\re$ and
$\rem$, but for all values of the shear parameter.
