We present a comprehensive macroscopic thermodynamic study of the quasi one dimensional 1D s 1 2 frustrated spin chain system linarite. Susceptibility, magnetization, specific heat, magnetocaloric effect, magnetostriction, and thermal expansion measurements were performed to characterize the magnetic phase diagram. In particular, for magnetic fields along the b axis five different magnetic regions have been detected, some of them exhibiting short range order effects. The experimental magnetic entropy and magnetization are compared to a theoretical modeling of these quantities using density matrix renormalization group DMRG and transfer matrix renormalization group TMRG approaches. Within the framework of a purely 1D isotropic model Hamiltonian, only a qualitative agreement between theory and the experimental data can be achieved. Instead, it is demonstrated that a significant symmetric anisotropic exchange of about 10 is necessary to account for the basic experimental observations, including the three dimensional 3D saturation field, and which in turn might stabilize a triatic three magnon multipolar phase