2 I have added an extreme example

The zeta function is real on the critical line only at the zeros and at Gram points, this is because zeta(1/2+it)=exp(-ivartheta(t)) Z(t).

At the Gram point g_k we have by definition vartheta(g_k)=pi k. so that zeta(1/2+ig_k) =(-1)^k Z(g_k).

Now a Gram point g_k is said a good Gram point if (-1)^k Z(g_k) >0. In other case it is said a bad Gram point.
Since it appear improbable a zero just at a Gram point. You are asking if there exists bad Gram points, there are plenty. The first few bad Gram points are g_126, g_134, g_195, g_211, ...

g_126 = 282.45472082346217461077

In fact it is proved there are infinite bad Gram points.

Also we may easily obtain large negative values. For example using data of T. Kotnik "Computational estimation of the order of zeta(1/2+it) Math of Comp. (2003) we easily locate the point t = grampoint(2601005843707) were we have

zeta(0.5+i t) = -119.6304321077241661374

This is easily confirmed in mpmath (or Mathematica) ( grampoint(2601005843707) = 669980906189.53552206792 ).

1

The zeta function is real on the critical line only at the zeros and at Gram points, this is because zeta(1/2+it)=exp(-ivartheta(t)) Z(t).

At the Gram point g_k we have by definition vartheta(g_k)=pi k. so that zeta(1/2+ig_k) =(-1)^k Z(g_k).

Now a Gram point g_k is said a good Gram point if (-1)^k Z(g_k) >0. In other case it is said a bad Gram point.
Since it appear improbable a zero just at a Gram point. You are asking if there exists bad Gram points, there are plenty. The first few bad Gram points are g_126, g_134, g_195, g_211, ...

g_126 = 282.45472082346217461077