We discuss the formalism of Iwasawa theory descent in the setting of the localized $K_1$-groups of Fukaya and Kato. We then prove interpolation formulas for the `leading terms' of the global Zeta isomorphisms that are associated to certain Tate motives and of the $p$-adic $L$-functions that are associated to certain critical motives.

11G40, 11R65, 19A31, 19B28

Zeta isomorphisms, Tate motives, $p$-adic $L$-functions, non-commutative Iwasawa theory, $K \sb 1$-groups