Thank you Kirill. But I don't understand the answer to second part. In your notation I am looking for value of f in neighbourhood of $0$. Is it that $f$ has a minimum near $-4.89$? up to what range are you considering the value of $x$, for $-4.89$ to be minimum?

@Stopple It seems $\gamma_0$(or $\gamma_1$) is irrelevant in 2) unless I provide the bound for $\theta$. It was a mistake of mine. Thank you for pointing it out. I wanted to understand the value of $\zeta(s)$ near $1/4 + \gamma_0/2$. The value of \theta may be assumed to vary between $[0, \delta_0]$, when $\delta_0\leq 2\pi$ (The bound $2\pi$ is from the context I am working on). Insted of an absolute(independent of \theta) lower bound of $\zeta(s)$, is it possible to get a lower bound in terms of $\theta$?

@Kirill can you please mention, how you calculated it. I find it difficult to do.

Onething to observe here that to construct $D_1$ and $D_2$ , we just need the cycle lenghts of $w_2w_1^{-1}$ and a representative for each cycle.

Still I did not get the proof. Here the problem is diagonal matrices do not commute with permutation matrices.

@TheMaskedAvenger Diagonal matrix means different elements in the diagonal are also allowed. For example, \begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix} is a diagonal matrix.