Personally, yes, functional requirements should be written as acceptance criteria within the issue. Best way to get the team to ensure they are all executed is to put them in the team's workflow. The team is using JIRA to track their work so put the acceptance criteria there as well. If you document them somewhere else, it may work as well but the team will then have information scattered around and may forget about some criteria before closing the issue.
Some people document their acceptance criteria in the description field of the issue (the wiki renderer allows checkmark icons). Other use third party add-on such as Checklist (shameless plug here ). The nice thing about these add-ons is that they nicely format the criteria and some actually allow you to prevent the team to resolve/close the issue before all the items in the list are checked.
Hope that helps
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