As you have asked about the list of centres available after registration for Law School Admission test (LSAT) in Ohio, I am mentioning them, check below for the details
You can register online for LSAT and select the following as the centres in Ohio,
Ada,
Ohio Northern University

Akron,
University of Akron

Cedarville,
Cedarville University

Cincinnati,
University of Cincinnati

Cleveland,
Case Western Reserve University School of Law

Columbus,
Ohio State University

Dayton,
University of Dayton

Oberlin,
Oberlin College

Oxford,
Miami University

Toledo,
The University of Toledo College of Law

Wooster,
The College of Wooster

Youngstown,
Youngstown State University