On the zeros of the Ramanujan tau-Dirichlet series in the critical strip

We describe computations which show that each of the first 12069 zeros of the Ramanujan τ -Dirichlet series of the form σ + it in the region 0 < t < 6397 is simple and lies on the line σ = 6. The failures of Gram’s law in this region are also noted. The first 5018 zeros and 2228 successive zeros beginning with the 20001st zero are also calculated. The distribution of the normalized spacing of the zeros is examined and it appears to be that of the eigenvalues of random matrices. These comptuations are done with a BerryKeating formula for the τ -Dirichlet series and evaluated using MathematicaTM.