How to calculate the Gain (in dB) given ω₁ and sampling time, T, of a discrete transfer function?
First of all, convert the Z-transform $H(z)$ into the Frequency response $H(e^{j\omega})$ by replacing $z$ with $e^{j\omega}$.

$$ \begin{align}
H(z) &amp;= (1.011) ~\frac{z-0.9756}{z-0.9975} \\ \\
&amp;= (1.011)~\frac{1-0.9756 ~z^{-1}}{1-0.9975~z^{-1}} \\ \\H(e^{j\omega}) &amp;= (1.011)~\frac{1-0.9756 ~e^{-j\omega}}{1-0.9975 ~e^{-j\omega}}\\ 
\end{align}
$$

Then find the discrete-time radian frequency $\omega$ from the given continous-time frequency $\Omega = 3 ~\text{rad\s}$ by the  sampling relation (assuming no aliasing):

$$ \Omega = 3 ~ \text{rad/s} \implies \omega = \Omega \times T_s = 3 \times 0.25 = \frac{3}{4} ~\text{rad/sample}$$

Then find the gain of the LTI system at the given frequency $\omega = 3/4$ :

$$ 
H(e^{j3/4}) = (1.011)\frac{1-0.9756 ~e^{-j3/4}}{1-0.9975 ~e^{-j3/4}}
$$

Gain is the magnitude: $|H(e^{j3/4})| = 1.0004$. And the gain in dB is given by:

$$ G_{db} ~= 20 \log_{10}( |H(e^{j3/4})| ) ~= 0.0035 ~~\text{dB} $$