Zeros of Zeta function and exact roots

Question

Are there exact roots to any of the Zeta zeroes? For example the first one 1/2 +14.134725I, is there a nice looking polynomial that has an exact solution? I would assume if there is an exact value, than there would also be a conjugate. The conjugate would be 1/2 -14.134725I. I imagine it would also include negatives. I've seen polynomials with exact zeta conjugates. Here is a polynomial with zeta conjugates in it. Sorry it's so big.

$${\frac {256\,{x}^{20}-640\,{x}^{18}+560\,{x}^{16}-200\,{x}^{14}+25\,{x }^{12}-{x}^{10}+25\,{x}^{8}-200\,{x}^{6}+560\,{x}^{4}-640\,{x}^{2}+256 }{{x}^{10}}}$$

output of nth root as input for Zeta function

1. 0.77196759e-1-40.855557*I
2. 0.77196759e-1+40.855557*I
3. 0.75855086e-1-4.7847003*I
4. 0.75855086e-1+4.7847003*I
5. 0.50739489e-1-.95183871*I
6. 0.50739489e-1+.95183871*I
7. 0.21253471e-1-.54700119*I
8. 0.21253471e-1+.54700119*I
9. -0.83307942e-1+0.40490429e-2*I
10. -0.83307942e-1-0.40490429e-2*I
11. -0.81509478e-1+0.34644023e-1*I
12. -0.81509478e-1-0.34644023e-1*I
13. -0.49293923e-1+.17966248*I
14. -0.49293923e-1-.17966248*I
15. -0.15358441e-1+.31869497*I
16. -0.15358441e-1-.31869497*I
17. 0.68554652e-1+1.8289288*I
18. 0.68554652e-1-1.8289288*I
19. -0.71850985e-1+0.91582220e-1*I
20. -0.71850985e-1-0.91582220e-1*I

Answer 1

In other words, you are asking us if the non-trivial zeroes of the Riemann $\zeta$ function are algebraic. The official answer to this question is that we don't know yet, i.e., it hasn't been proven either one way or the other. However, as far as educated guesses are concerned, no mathematician expects this to be the case; more to the point, they are expected to be transcendental. Indeed, the higher the numerical precision with which we compute the root in question, the larger the polynomial degree, and the greater its coefficients.