Continuous prime systems satisfying N(x)=c(x-1)+1

Hilberdink showed that a continuous prime system for which there exists constant $A$ such the function $N(x)-Ax$ is periodic satisfies $N(x)=c(x-1)+1$. He further $c_0>2$, of this form if and only $c\leq c_0$. Here we determine $c_0$ numerically to be $1.25479\cdot 10^{19}\pm2\cdot 10^{14}$. To do so compute representation twisted exponential as sum over roots Riemann zeta function. We then give explicit bounds error obtained when restricting occurring finite number zeros.