TITLE: Negative values of Riemann zeta function on the critical line.
QUESTION [8 upvotes]: From parametric plots of $\zeta \left( \frac{1}{2} + it \right)$ it seems to be the case that:
(1) except for $\zeta(\frac{1}{2})$ the Riemann zeta function does not attain any negative real value on the critical line.
(2) the curve $(t, \zeta(1/2+it))$ is dense in the complex plane.
Are these statements known to be false, if not, is there any proof affirming them?

REPLY [7 votes]: Update, some recent information on (1): 
Kalpokas, Korolev, Steuding recently released a preprint showing that $\zeta(1/2 + it)$ takes aribtrarily large positive and negative (real) values; and also show analog statements for the other lines through the origin, that is positive and negative (real) values of arbitary says of $e^{-i \phi} \zeta(1/2 + it)$ for any $\phi$. The paper contains also more quantitative results along these lines (cf. in particular Corollary 3 and the preceeding discussion). 

Since (1) already received several answers, I expand and upgrade the comments on (2):
Yes, indeed it is conjectured, but unproved, that $\zeta(1/2 + i t)$ for $t \in \mathbb{R}$ is dense in the complex plane. [Side note: It is well-known that this is so for the lines $\sigma +it$ with $1/2 < \sigma < 1$.]
It seems that this conjecture was first formulated by Ramachandra (Durham, 1979), however only appeared in print in the second edition of Titchmarsh's book (note's by Heath-Brown), see the articles below for details.  
There is very recent work on this problem due to Delbaen, Kowalski, and Nikeghbali. 
See in particular this preprint by the latter two and this by all three. 
Among others: in the former, they show how this result would follow "from a suitable version of the Keating--Snaith moment conjectures"; 
in the latter, they propose a refinement of the density conjecture, a quantitative version (see Conj. 1, in Sec. 3.9).

REPLY [7 votes]: 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 ).