What is the ruling on slaughtering a pigeon or a jerboa by hand (wringing its neck - twisting its neck) without a sharp instrument after saying "Allahu Akbar"?

It is not permissible to eat wild animals unless they are slaughtered according to Islamic law (dhakah shar'iyyah), which is performed with a sharp instrument that severs the esophagus and the two jugular veins by a Muslim or a discerning, sane person from the People of the Book. Decapitating a pigeon or a gerbil by hand is not considered a valid Islamic slaughter. Whatever is killed in this manner is considered carrion, and it is not permissible to eat it, due to the Almighty's saying: "Forbidden to you are carrion."