The classical Coates-Sinnott Conjecture and its refinements predict the dee relationship between the special values of L-functions and the structure of the étale cohomology groups attached to number fields. In this talk, we aim to delve deeper along this direction to propose what we call the “Higher Coates-Sinnott Conjectures" which reveal more information about these two types of important arithmetic objects. We introduce the conjectures we formulate and our work towards them. This is joint work with C. Popescu.