Let $P(X)$ be a polynomial of degree $N$ with complex coefficients and $d\sb 1$, $d\sb 2$ two complex numbers with real part greater than $-1$. By using certain properties of Bessel functions, the authors prove an explicit formula in terms of Hurwitz zeta functions for the Dirichlet series $L(s)=\sum\sp \infty\sb {n=1} P(n)(n+d\sb 1)\sp {-s} (n+d\sb 2)\sp {-s}$, which gives the meromorphic continuation of $L(s)$ to the whole complex plane, with at most simple poles at $s=(N+1-k)/2\ (k=0,1,\cdots)$. Their result includes, as a special case, a function-theoretic proof of the meromorphic continuation of the Minakshisundaram zeta function of the 3-dimensional sphere.