  Begin with the Hasse-Weil zeta-function of a smooth projective variety over
the rational numbers. Replace the variety with a finite CW-complex, replace
etale cohomology with complex K-theory $KU^*$, and replace the $p$-Frobenius
operator with the $p$th Adams operation on $K$-theory. This simple idea yields
a kind of "$KU$-local zeta-function" of a finite CW-complex. For a wide range
of finite CW-complexes $X$ with torsion-free $K$-theory, we show that this
zeta-function admits analytic continuation to a meromorphic function on the
complex plane, with a nice functional equation, and whose special values in the
left 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 $KU$-local
zeta-function, one which is suited to finite CW-complexes $X$ with nontrivial
torsion in their $K$-theory. This more sophisticated $KU$-local zeta-function
involves a product of $L$-functions of complex representations of the torsion
subgroup of $KU^0(X)$, similar to how the Dedekind zeta-function of a number
field factors as a product of Artin $L$-functions of complex representations of
the Galois group. For a wide range of such finite CW-complexes $X$, we prove
analytic continuation, and we show that the special values in the left
half-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.
