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Begin with the Hasse-Weil zeta-function of a smooth projective variety overthe rational numbers. Replace the variety with a finite CW-complex, replaceetale cohomology with complex K-theory $KU^*$, and replace the $p$-Frobeniusoperator with the $p$th Adams operation on $K$-theory. This simple idea yieldsa kind of "$KU$-local zeta-function" of a finite CW-complex. For a wide rangeof finite CW-complexes $X$ with torsion-free $K$-theory, we show that thiszeta-function admits analytic continuation to a meromorphic function on thecomplex plane, with a nice functional equation, and whose special values in theleft half-plane recover the $KU$-local stable homotopy groups of $X$ away from$2$.  We then consider a more general and sophisticated version of the $KU$-localzeta-function, one which is suited to finite CW-complexes $X$ with nontrivialtorsion in their $K$-theory. This more sophisticated $KU$-local zeta-functioninvolves a product of $L$-functions of complex representations of the torsionsubgroup of $KU^0(X)$, similar to how the Dedekind zeta-function of a numberfield factors as a product of Artin $L$-functions of complex representations ofthe Galois group. For a wide range of such finite CW-complexes $X$, we proveanalytic continuation, and we show that the special values in the lefthalf-plane recover the $KU$-local stable homotopy groups of $X$ away from $2$if and only if the skeletal filtration on the torsion subgroup of $KU^0(X)$splits completely.

KU-local zeta-functions of finite CW-complexes