11 Number theory -- 11M Zeta and $L$-functions: analytic theory 11M41 Other Dirichlet series and zeta functions
58 Global analysis, analysis on manifolds -- 58G Partial differential equations on manifolds; differential operators 58G26 Determinants and determinant bundles
Let $P(x)$ be a polynomial with $P(0)=0$, $a$ and $b$ non-negative rational numbers, and consider the Dirichlet series $$L(s)=\sum_{n=1}^{\infty}\frac{P(n)}{(n+a)\sp s (n+b)\sp s }.$$ In a previous paper [Proc. Amer. Math. Soc. {\bf 121} (1994), no.~1, 33--37; MR1179586 (94g:11066)], the authors obtained a representation of $L(s)$ in terms of Hurwitz zeta-functions. In the present paper, a functional equation of the Riemann type is established for $L(s)$ under certain assumptions, and the result is applied to Minakshisundaram-Pleijel zeta-functions of the real spheres and real and complex projective spaces.