Abstract: We discuss a conjectural description of leading Taylor coefficients of Zeta functions of arithmetic schemes in terms of Weil-etale cohomology of motivic complexes. For varieties over finite fields this goes back to Milne, Lichtenbaum and Geisser, and for schemes of characteristic zero it amounts to more geometric and global reformulation of the Tamagawa number conjecture of Bloch, Kato, Fontaine and Perrin-Riou. We discuss some partial constructions of such a Weil-etale cohomology for s=0 (joint work with Morin) and for all n for the Dedekind Zeta function.