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In this paper, we show that any polynomial of zeta or $L$-functions with someconditions has infinitely many complex zeros off the critical line. Thisgeneral result has abundant applications. By using the main result, we provethat the zeta-functions associated to symmetric matrices treated by Ibukiyamaand Saito, certain spectral zeta-functions and the Euler-Zagier multiplezeta-functions have infinitely many complex zeros off the critical line.Moreover, we show that the Lindel\"of hypothesis for the Riemann zeta-functionis equivalent to the Lindel\"of hypothesis for zeta-functions mentioned abovedespite of the existence of the zeros off the critical line. Next we prove thatthe Barnes multiple zeta-functions associated to rational or transcendentalparameters have infinitely many zeros off the critical line. By using thisfact, we show that the Shintani multiple zeta-functions have infinitely manycomplex zeros under some conditions. As corollaries, we show that the Mordellmultiple zeta-functions, the Euler-Zagier-Hurwitz type of multiplezeta-functions and the Witten multiple zeta-functions have infinitely manycomplex zeros off the critical line.

Any non-monomial polynomial of the Riemann zeta-function has complex zeros off the critical line