On the Bautforum there's actually a thread going on about the stability of the Lagrangian points of the Earth and Moon .  
http://www.bautforum.com/questions-answers/74989-orbit-question-3.html
Theres a reference to the Neil Cornish paper which calculates the Lagrangian points  
http://map.gsfc.nasa.gov/ContentMedia/lagrange.pdf
 
With the formulas in this paper one can calculate the position and velocities of the bodies residing in the L1,L2,L3 points  : ( referenced to the mass center of Earth/Moon )  
"Earth", -4665880.79,  -12.455,  
"Moon", 379334119.21, 1012.617
"MoonL1", 321375133.17, 857.90
"MoonL2", 443782166.58, 1184.659
"MoonL3", -385944079.29,  -1030.262
 
Using the following data ( as Tony did )  
For Earth: 5.9736E+24 kg
For Moon: 7.34764E+22
G: 6.6725985E-11
R =384000000 m  
 
All of the points are known to be unstable. But how "unstable" ?  
Tony is correct as he says the L3 point is rather stable . (L3 being the point opposite to the moon ) .  
To find out the degree of instability the above data were input in the Picard Integrator and ran for 1 year . Screenshots were made every  1/100 year.  
One can see how all the test masses leave their original position after a short time .  
L3 keeps its position for a rather long time . L1 seems to become a moon of our moon .
I'll try to post a rotating frame also which is far more expressive ...  
( hint : click on the shortcut to open a separate frame )  
 
 

I believe the formulas in Cornish's paper are only approximations that don't yield good results when the mass of the secondary is not insignificant compared to the primary
 
I made a javascript calculator to compute the L points:
http://orbitsimulator.com/formulas/LagrangePointFinder.html

In order to get some feeling with the position of the L1 and L2 points of a two body system I made the following animation ...
X-axis shows at zero the mass center . X=1 is the position of the secondary .  
The Y-axis represents the value of u²(....) = .... in the Cornish paper referenced above .  
The L1 point is located at the intersection of the blue line with 0 ; the L2 point is positioned at the intersection of the pink line .  
The parameter M2/(M1+M2) is increased in steps of 0.02 . So the animation starts at zero mass of the secondary .  
Animation stops at the point where the two masses are equal .  
The positon of the L2 point tends to be around 1.2 R , whereas L1 approches to the center of mass when M2=M1 . This is logical .  
However L2 seems to be frozen at 1.2 R in this case , which I didn't expect . So , even in this case there seems to be a L2 point ?  

Quote from Tony on 06/21/08 at 13:37:20:

I like this graph too ... Tells a lot more then a formula .
Any idea what the L2 at M2=M1 might be at x/R =1.2 , ie beyond the secondary ?  

Quote from Dan on 06/21/08 at 19:14:00:

 
To the best of my knowledge, every field works.  Velocity remains blank until you fill in a value for secondary mass.  But after all 3 inputs are filled, you should have values in all the outputs.  So if you're still not getting a value let me know.  It may be a browser issue.  What browser are you using?  I've only tested it against IE and Firefox.

Applying the Cornish equation in reference above to varying mass ratios of the secondary to the primary body I got the following animation .  
The graph shows the position of the Lagrangian points to the primary and secondary ( blue dots ) for increasing mass ratios M2/M1.  
It's interesting to see that the L4/L5 points don't change at all , L1 and L2 change a lot , while L3 is not so much affected by the varying mass ratio.