Let
P
(
x
)
=
∏
j
=
2
k
(
x
+
δ
j
)
P(x) = \prod \nolimits _{j = 2}^k {(x + {\delta _j})}
be a polynomial with real coefficients and
Re
δ
j
>
−
1
(
j
=
1
,
…
,
k
)
\operatorname {Re} {\delta _j} > - 1(j = 1, \ldots ,k)
. Define the zeta function
Z
p
(
s
)
{Z_p}(s)
associated with the polynomial
P
(
x
)
P(x)
as
\[
Z
P
(
s
)
=
∑
n
=
1
∞
1
P
(
n
)
s
,
Re
s
>
1
/
k
.
{Z_P}(s) = \sum \limits _{n = 1}^\infty {\frac {1}{{P{{(n)}^s}}}} ,\operatorname {Re} s > 1/k.
\]
Z
P
(
s
)
Z_P(s)
is holomorphic for
Re
s
>
1
/
k
\operatorname {Re} s > 1/k
and it has an analytic continuation in the whole complex
s
s
-plane with only possible simple poles at
s
=
j
/
k
(
j
=
1
,
0
,
−
1
,
−
2
,
−
3
,
…
)
s = j/k(j = 1,0, - 1, - 2, - 3, \ldots )
other than nonpositive integers. In this paper, we shall obtain the explicit value of
Z
P
(
−
m
)
{Z_P}( - m)
for any non-negative integer
m
m
, the asymptotic formula of
Z
P
(
s
)
{Z_P}(s)
at
s
=
1
/
k
s = 1/k
, the value
Z
P
′
(
0
)
{Z’_P}(0)
and its application to the determinants of elliptic operators.