It doesn't multiply your production or take your production/Dungeons into account whatsoever. What it does is multiply the actual recruits that you have. If you have seven and play it, after the cost of the card you will have 5, so it will grant you +5 and then also give you your production as usual. Optimally, you want to play it when you have 12 recruits so it will double the 10 that you have once it's played.
In your case, the number of recruits you had once the cost was subtracted was 0. You gained +0 recruits by card effect, and the system doesn't display that as it doesn't register a change as having occurred. You gained your regular production after that setting you at 4.
To use your math formula above, it would be ((2 on hand - 2 cost)x2 + 4 dun).