The Riemann's Functional Equation as on wikipedia here:
describes the zeta function for all complex plane. But all integers are also in the complex plane. Thus,we can substitute for s=1 on the Left Hand Side of above equation.
If we substitute on the Left hand side s=1,
the Right hand side becomes:
= 21 * π0 * sin(π/2) * Factorial(0) * E(0)
= 2 * 1 * (1) * 1 * (-1/2)
= -1
Thus, E(1)=-1
Is this correct derviation or is there a problem in this with respect to meromorphic function theory related issue?
$$\zeta(s) \propto \Gamma(1-s) \implies \zeta(1) \propto \Gamma(0)$$
$\Gamma(z)$ does not exist for $z=0,-1,-2,\cdots$. This can be shown easily from the recursion property: $\Gamma(z+1)=z \Gamma(z)$. Let $z=0$ and solve for $\Gamma(0)$ to find a problem.
Note that $\Gamma(z+1) = z!$ when $z$ is a nonnegative integer, by the way. So if anything, $\Gamma(0) = (-1)!$ (were that to be defined).
In short, your derivation is incorrect and in turn pokes no holes in existing theory.