Ohio Rev. Code Ann. § 2725.03

If a person restrained of his liberty is an inmate of a state benevolent or correctional institution, the location of which is fixed by statute and at the time is in the custody of the officers of the institution, no court or judge other than the courts or judges of the county in which the institution is located has jurisdiction to issue or determine a writ of habeas corpus for his production or discharge. Any writ issued by a court or judge of another county to an officer or person in charge at the state institution to compel the production or discharge of an inmate thereof is void.