Inertial confinement fusion (ICF) hydrodynamic codes often ignore the effects of viscosity though recent research indicates plasma viscosity and mixing by classical transport processes may have a substantial impact on implosion dynamics. A Lagrangian hydrodynamic code in one-dimensional spherical geometry with plasma viscosity and mass transport, and including a three temperature model for ions, electrons, and radiation treated in a gray radiation diffusion approximation, is used to study differences between ICF implosions with and without plasma viscosity and to examine the role of artificial viscosity in a Lagrangian implosion simulation. It was found that plasma viscosity has substantial impacts on ICF shock dynamics characterized by shock burn timing, maximum burn temperatures, fuel compression, and time history of neutron production rates. Plasma viscosity reduces the need for artificial viscosity to maintain numerical stability in the Lagrangian formulation and this study suggests that artificial viscosity may provide an unphysical stability in implosion simulations.