Clarification Regarding Changing the Variable
The implication seems a little bit wrong, but that is okay.
When dealing with the functions in the frequency domain, let’s discuss it over the function $X(j\omega)$, when shifting or scaling the function on the frequency axis, the operations are done on the angular frequency $\omega$ or the frequency $f$.
On the frequency axis, there is no imaginary number, that is, the frequency axis is full of real numbers. Therefore, calling a point on the frequency axis as e.g. $j2$ Hz is incorrect.
When I do calculations on the paper, I prefer using $\omega$ instead of $j\omega$ when defining the variable of the Fourier transform of a function e.g. $X(\omega)$. However, in the case of transferring a Fourier transform into the Laplace transform under specific conditions, $j\omega$ notation aids in preventing confusion of where to put the $s$ variable as in this process, we do assignment $s = j\omega$.
I've changed the given function a little bit i.e. I've taken the absolute value of the angular frequency $\omega$ in the power of the exponential in order to illustrate what happens better.
Okay, along with these knowledge, let’s focus on your function. The amplitude spectrum of the function $X(j\omega)$ is given below:

As it is noted, the mathematical explanation of the function is;
$$X(j\omega)=\begin{cases}e^{-2|\omega|},&amp;\text{if $|\omega|&lt;2\pi$}\\0,&amp;\text{otherwise}\end{cases}$$
So, let’s shift the function $X(j\omega)$ by $\Omega$ to the right on the frequency axis as I’ve assumed $\Omega&gt;0$. This shifting process, which is a frequency shifting, is done in the same way as time shifting. The resultant shifted function $X(j(\omega - \Omega))$ will be in the figure below:

So, the resultant mathematical expression becomes;
$$X(j(\omega - \Omega))=\begin{cases}e^{-2(|{\omega - \Omega}|)},&amp;\text{if $|{\omega - \Omega}|&lt;2\pi$}\\0,&amp;\text{otherwise}\end{cases}$$
The expression $|{\omega - \Omega}|&lt;2\pi$ can be represented comprehensively as ${{-2\pi}+{\Omega}}&lt;{\omega}&lt;{{2\pi}+{\Omega}}$.