We report on work, joint with Morin, that gives a conjectural description of leading Taylor coefficients of Zeta functions of arithmetic schemes in terms of volumes of certain Weil-etale cohomology groups of motivic complexes. Such a description was given by Milne, Lichtenbaum and Geisser for varieties over finite fields and was begun by Lichtenbaum for the Dedekind Zeta function at s=0. Our work covers arbitrary regular, projective arithmetic schemes at any integer argument and is compatible with the Tamagawa number conjecture of Bloch, Kato, Fontaine and Perin-Riou.