  We perform numerical experiments to study the shear dynamo problem where we
look for the growth of large--scale magnetic field due to non--helical stirring
at small scales in a background linear shear flow, in previously unexplored
parameter regimes. We demonstrate the large--scale dynamo action in the limit
when the fluid Reynolds number (${\rm Re}$) is below unity whereas the magnetic
Reynolds number (${\rm Rm}$) is above unity; the exponential growth rate scales
linearly with shear, which is consistent with earlier numerical works. The
limit of low ${\rm Re}$ is particularly interesting, as seeing the dynamo
action in this limit would provide enough motivation for further theoretical
investigations, which may focus the attention to this analytically more
tractable limit of ${\rm Re} < 1$ as compared to more formidable limit of ${\rm
Re} > 1$. We also perform simulations in the regimes when, (i) both (${\rm
Re}$, ${\rm Rm}$) $< 1$; (ii) ${\rm Re} > 1$ & ${\rm Rm} < 1$, and compute all
components of the turbulent transport coefficients ($\alpha_{ij}$ and
$\eta_{ij}$) using the test--field method. A reasonably good agreement is seen
between our results and the results of earlier analytical works (Sridhar &
Singh 2010; Singh & Sridhar 2011) in the similar parameter regimes.
