This paper offers a breakthrough in proving the veracity of original Riemann hypothesis, and extends the validity of its method to include the cases of the Dedekind zeta functions, the Hecke L-functions hence the Artin L-functions, and the Selberg class.First we parametrize the Riemann surface $mathbf{S}$ of $log$-function, with which we first shrink the scale of each chosen parameter for which it depends on the chosen natural number $Q_{N_{0}}$ which is a chosen common multiple of all the denominators which are derived from a pre-set choice of rational numbers which approximate the values $log(k+1)$ with the integers $k$ in $0leq kleq N$.Then in (1.7) we define the mapping $-Q_{N_{0}}log(.)$ to pull the truncated Dirichlet $eta$-function $f_{N}(s)$ back to be re-defined on $mathbf{S}$, after that we shrink all the points to have their absolute values are all less than $1$ and closer to $1$. We apply the Euler transformation to the alternative series of Dirichlet $eta$-functions $f(s)$ which are defined in (1.4), then we build up the locally uniform approximation of Theorem 4.7 for $f(s)$ which are established on any given compact subset contained in the right half complex plane.In the second part we define the functions $phi(s)$ which are formulated in (6.1) then by specific property of the functions $phi(s)$, we have the similar asymptotics Theorem 6.5 as those of Theorem 4.7 to obtain the result of Theorem 6.8.And with the locally uniform estimation Lemma 5.10, finally in Theorem 5.9 and Theorem 6.9 we employ Theorem 5.8 and Theorem 6.8 to solve problems of Riemann hypothesis for the Dedekind zeta functions, the Hecke L-functions, the Artin L-functions, and the Selberg class for which all of their nontrivial zeros are contained in the vertical line $Re(s)=1/2$.Finally for the $gamma(s)$-factor of each Dirichlet series $D(s)$ which is formulated in (1.4), then by Theorem 6.9 it has neither zeros nor poles contained in the critical strip $0<Re(s)<1$ and the non-existence of Siegel's zeros for such Dirichlet series $D(s)$ is confirmed.