Let
P
(
X
)
P(X)
be a polynomial of degree N with complex coefficients and
d
1
,
d
2
{d_1},{d_2}
two complex numbers with real part greater then
−
1
-1
. Consider the Dirichlet series associated with the triple
(
P
(
X
)
,
d
1
,
d
2
)
(P(X),{d_1},{d_2})
\[
L
(
s
)
=
∑
n
=
1
∞
P
(
n
)
(
n
+
d
1
)
s
(
n
+
d
2
)
s
.
L(s) = \sum \limits _{n = 1}^\infty {\frac {{P(n)}}{{{{(n + {d_1})}^s}{{(n + {d_2})}^s}}}.}
\]
In this paper we get an explicit formula for
L
(
s
)
L(s)
in terms of special functions which gives meromorphic continuation of
L
(
s
)
L(s)
with at most simple poles at
s
=
(
N
+
1
−
k
)
/
2
,
k
=
0
,
1
,
…
s = (N + 1 - k)/2,k = 0,1, \ldots
Finally we apply our explicit formula to Minakshisundaram’s zeta function of the three-dimensional sphere.