You should not use Xbox Live while exploited to avoid a console ban; it is best used with an offline profile.
Released as a breakthrough hypervisor exploit, allows for homebrew execution on every Xbox 360 model, including the previously "unhackable" Winchester motherboards. Xbox 360 Jailbreak Software Download
Before proceeding with any and the associated hardware work, consider these real-world consequences. You should not use Xbox Live while exploited
If you are serious about jailbreaking your Xbox 360, here is the software you need for each method. Xbox 360 Jailbreak Software Download
On Xbox One/Series, you can pay a $20 fee to unlock Developer Mode and run retroarch and emulators. This is legal, safe, and supported.