In the set of non-trivial zeros of the Riemann zeta function, there are some numbers that are unusually close to each other. These are called Lehmer pairs, the first of which appears at the 6709th and 6710th zeros of the zeta function. A question that might come to mind after knowing this fact is: are there any Lehmer triplets? i.e. three consecutive zeros that are unusually close to each other.
I tried asking google and consulted ChatGPT without any luck. A combination of different keywords returns some results such as this one that are completely irrelevant. Maybe they do exist, but are called something else. My question(s) are:
Have we found any set of three or more consecutive zeros that are very close to each other?
Are there any research articles about this specific question?
If the answer to the first question is negative, do we know that they exist or not? Does their existence violate some known results?