motives – Values of cohomology theory on a point

It is a well-known fact that in algebraic topology, generalized cohomology theories are determined by their values on the point. I was wondering whether anything similar to that is or expected to be true in algebraic geometry? More precisely I had the following question in my mind:

  • For a Bloch-Ogus cohomology theory on the category of smooth $k$ schemes with the big Zariski site, if we know that cohomologies of $text{Spec}(k)$ at all dimensions and weights are trivial, does it imply that the cohomology theory is trivial?

Bloch-Ogus cohomology theory is a functor from $Sm_k$ to abelian groups that has two grading $h^i(X,j)$ similar to the motivic cohomology. These functors are supposed to satisfy certain properties like homotopy invariance, purity, transfer under finite maps and maybe some other extra properties (like Poincare duality).