1. Problem statement: The function $\zeta(s)=\sum_{n=1}^\infty n^{-s}$ is defined for $\Re(s)>1$ and admits an analytic continuation to $\mathbb{C}\setminus\{1\}$; the question asks whether the non-trivial zeros of this continuation all have the form $\tfrac{1}{2}+ix$ with $x\in\mathbb{R}$. 2. Definition and analytic continuation: For $\Re(s)>1$ we have $\zeta(s)=\sum_{n=1}^\infty n^{-s}$ and this Dirichlet series defines a holomorphic function in that half-plane. 3. Completed function and functional equation: Define the entire completed function by $$\xi(s)=\tfrac{1}{2}s(s-1)\pi^{-s/2}\Gamma\left(\frac{s}{2}\right)\zeta(s)$$ and note that it satisfies the symmetry $$\xi(s)=\xi(1-s)$$ which is equivalent to the classical functional equation for $\zeta(s)$. 4. Symmetry of zeros: From $\xi(s)=\xi(1-s)$ we deduce that if $\xi(s_0)=0$ then $\xi(1-s_0)=0$, so zeros of $\xi$ are symmetric with respect to the vertical line $\Re(s)=\tfrac{1}{2}$. 5. Trivial zeros versus non-trivial zeros: The prefactor $\tfrac{1}{2}s(s-1)\Gamma(\tfrac{s}{2})$ forces zeros at the negative even integers $s=-2,-4,-6,\dots$, which are the trivial zeros of $\zeta(s)$ and are well understood. 6. Location of non-trivial zeros: The remaining zeros of $\zeta(s)$ (equivalently zeros of $\xi(s)$ not coming from the prefactor) lie in the critical strip $0<\Re(s)<1$ and are called non-trivial zeros; by the functional equation and complex conjugation these zeros occur in symmetric patterns about $\Re(s)=\tfrac{1}{2}$ and about the real axis. 7. The Riemann Hypothesis: The assertion that every non-trivial zero has real part exactly $\tfrac{1}{2}$ is the Riemann Hypothesis, and this statement is a famous open problem that has not been proven. 8. Known partial results: There are many deep partial results: Hardy proved infinitely many zeros lie on the critical line $\Re(s)=\tfrac{1}{2}$, extensive numerical verifications find billions of zeros on the line to very large heights, and analytic results give zero-density estimates and zero-free regions that constrain where counterexamples could lie. 9. Conclusion and answer to the question: One cannot currently show that all non-trivial zeros are of the form $\tfrac{1}{2}+ix$ because that is exactly the unresolved Riemann Hypothesis; what can be shown unconditionally is symmetry about the line $\Re(s)=\tfrac{1}{2}$ and confinement of non-trivial zeros to the critical strip, but not that the entire zero set lies on the line unless the Riemann Hypothesis is assumed or proven. 10. Recommendation for further study: For a rigorous, detailed development consult standard texts on the Riemann zeta function and analytic number theory such as Titchmarsh's or Edwards' treatments of $\zeta(s)$ and its zeros.