The General license is more complex than your view Jomfli. In order to kill birds under the General license you must be able to show that it was to either: Prevent serious damage to livestock, foodstuffs for livestock, crops, vegetables, fruit, growing timber, fisheries or inland waters, and/or
Prevent the spread of disease.
You should also have tried scaring and or proofing and shown that they are impractical before you move onto trapping or shooting.
As I said before it is unlike that a person in a domestic residence will comply. The problem then is that if your actions are reported you would then need to prove the above points. If you fail then you fall under the Wildlife and Countryside act 1981 which protects all UK birds from being killed unless allowed under other laws such as the General License above or in season, game birds. Pigeons are not game birds and cannot be shot soley for the pot. Fines are hefty and can be in the region of £5000