Lagrange Point and orbital velocity

[_Stew_]

When considering the Lagrange point 1 in the Sun/Earth system, Does Lagrange Point 1 have any orbital velocity around Earth?

I suspect a body at L1 has no orbital velocity around earth.
But consider the Earth's position 6months later when it is opposite the sun. The Lagrange point would also have moved to the opposite side of the earth. This would suggest the orbital period of a body at L1 around the Earth would be one year.

So back to my original question:
Does Lagrange Point 1 have any orbital velocity around Earth?

 

[Bandersnatch]

I suspect a body at L1 has no orbital velocity around earth.

You have not supplied any reasons for your suspicion, while supplying a good reason to the contrary in the following two sentences.
As seen from Earth(disregarding daily rotation), the body at L1 will have indeed moved across the celestial sphere at angular velocity equal to that of the Sun - 360deg/sidereal year.

There are, however, issues with using "in orbit around Earth" when talking about a 3-body system. It's similar to saying that the Sun has got some orbital velocity around Earth - it's not entirely without merit(vide: common barycentre), but not entirely correct either.

 

[_Stew_]

I suspect a body at L1 has no orbital velocity around earth.
But consider

I used the word “but” to indicate I was talking about something conflicting with that statement.
The reason I suspected that the orbital velocity of the L1 body around Earth was zero is a bit harder to explain. Which is why I didn’t put it in, but I will now.
When trying to find Lagrange point 1, many calculations/proofs I have seen use 3 forces:
1) The gravitational force between the Sun and the L1 body
2) The centripetal force acting at L1 around the Sun
3) The gravitational force between the Earth and the L1 body
I was thinking that there should be another force, the centripetal force acting at L1 from the Earth?
It must be that introducing this new force is illogical for some reason OR the centripetal force acting at L1 from Earth is zero, making the orbital velocity of the L1 body around Earth = 0.
In light of what you said:

There are, however, issues with using "in orbit around Earth" when talking about a 3-body system. It's similar to saying that the Sun has got some orbital velocity around Earth

I’m guessing that it is illogical to consider L1’s orbit around earth