In Louisiana, you should attempt to obtain a delayed birth certificate administratively before you attempt the court route. The Vital Records Registry will refer you to court if you can not meet the administrative requirements.
You should collect as much early information as you possibly can. You should try to present - -- - copies of early school records, early immunization records, early medical records, an early census record for your family group, your family bible, affidavits from folks who were at least ten years old when you were born who can attest to the fact that they were personally aware of your birth; selective service records, baptismal records, marriage license & application. The Vital Records Registry will give you a list of the types of records you should present and how much weight they will be accorded. If the VRR can not issue a delayed b.c. administratively they will advise you to go to Court. Common problems are that the early documents have different names or inconsistent birth dates.
At the point when you have to go to Court you would then file a Petition for a Delayed Birth Certificate. Most of your work will be done since you have already collected all of the documents.