  A new method is devised for calculating the Igusa local zeta function $Z_f$
of a polynomial $f(x_1,\dots,x_n)$ over a $p$-adic field. This involves a new
kind of generating function $G_f$ that is the projective limit of a family of
generating functions, and contains more data than $Z_f$. This $G_f$ resides in
an algebra whose structure is naturally compatible with operations on the
underlying polynomials, facilitating calculation of local zeta functions. This
new technique is used to expand significantly the set of quadratic polynomials
whose local zeta functions have been calculated explicitly. Local zeta
functions for arbitrary quadratic polynomials over $p$-adic fields with $p$ odd
are presented, as well as for polynomials over unramified $2$-adic fields of
the form $Q+L$ where $Q$ is a quadratic form and $L$ is a linear form where $Q$
and $L$ have disjoint variables. For a quadratic form over an arbitrary
$p$-adic field with odd $p$, this new technique makes clear precisely which of
the three candidate poles are actual poles.
