  Let $\mathbb{F}_{q}$ denote the finite field of order $q$ (a power of a prime
$p$). We study the $p$-adic valuations for zeros of $L$-functions associated
with exponential sums of the following family of Laurent polynomials
f(x_1,x_2,...,x_{n+1})=a_1x_{n+1}(x_1+{1\over
x_1})+...+a_{n}x_{n+1}(x_{n}+{1\over x_{n}})+a_{n+1}x_{n+1}+{1\over x_{n+1}}
where $a_i\in \mathbb{F}_{q}^*,\,i=1,2,...,n+1$. When n=2, the estimate of the
associated exponential sum appears in Iwaniec's work, and Adolphson and Sperber
gave complex absolute values for zeros of the corresponding $L$-function. Using
the decomposition theory of Wan, we determine the generic Newton polygon
($q$-adic values of the reciprocal zeros) of the $L$-function. Working on the
chain level version of Dwork's trace formula and using Wan's decomposition
theory, we are able to give an explicit Hasse polynomial for the generic Newton
polygon in low dimensions, i.e., $n\leq 3$.
