Question on finding a Riemann Zeta function zero for high values of s

Let s = 1/2 +  1370919909931995308226.68016095...

This value of s is close to a zero.  I am trying to refine this value to 
38 places precision.

However even using precision = 400,000 digits, I still cannot find 
zeta(s) I keep getting this error:

eta: precision too low in mpcosm1.

Are there any short cuts here?

I was looking at https://www-users.cse.umn.edu/~odlyzko/zeta_tables/zeros5

In particular the zeta zeroes are close together on this part of the 
number line.

My curiosity was driven by Lehmer's Phenomena where two zeroes are close 
together, but looking at Odlyzko's table, this appears to be a miscue, 
the higher up a person goes on the critical line real(s) = 1/2 the 
zeroes do pack closer and closer together as can be seen here.

It would be nice to be able to calculate zeta(s) for values given in his 
10^22 table and refine it to 38 digits accuracy.

Randall