N.H. Rev. Stat. Ann. § 630:6

The punishment of death shall be inflicted within the walls or yard of the state prison. The sheriff of the county in which the person was convicted, and 2 of his deputies, shall be present, unless prevented by unavoidable casualty. He shall request the presence of the attorney general or county attorney, clerk of the court and a surgeon, and may admit other reputable citizens not exceeding 12, the relations of the convict, his counsel and such priest or clergyman as he may desire, and no others.