Zeros of Dirichlet series

We are concerned here with Dirichlet series f(s)=1+∑n=2∞c(n)ns which satisfy a function equation similar to that of the Riemann zeta function, typically of the form f(s)=2sq1/2−sπs−1Γ(1−s)(sinπ2(s+κ))f(1−s), but for which the Riemann hypothesis is false. Indeed we show that the zeros of such functions are ubiquitous in the complex plane.