Odlyzko has computed a data set listing more than $10^9$ successive Riemann zeros, starting at a zero number beyond $10^{23}$. The data set relates to random matrix theory since, according to the Montgomery-Odlyzko law, the statistical properties of the large Riemann zeros agree with the statistical properties of the eigenvalues of large random Hermitian matrices. The statistical quality of this data set draws attention to correction terms to leading order predictions, both in the context of the Riemann zeros, and in random matrix theory. We give results on this theme, with the end result being that we are able to exhibit integrable functional forms in data.