Lorentz transformations of fields evaluated at a point

I'm am sure that I must be missing something very simple, so apologies in advance.

Considering the Lorentz transformation $\Lambda$ of a spinor fields, for the plane-wave solution $u(p)$, I cannot for the life of me agree why

(1) $$u^s(\Lambda^{-1} {p'}) = \Lambda_{\frac{1}{2}} u^s(p')$$

where

$$p' = \Lambda p$$

This is in Peskin & Schroeder, pg 59, just above equation (3.110).

I have tried to get this a dozen times, to no avail.

I know that, for a scalar field, under a Lorentz transformation $\Lambda$ we get, as per Peskin & Schroeder, pg 36, equation (3.2)

$$\phi(x) \rightarrow \Lambda \phi(x) = \phi'(x) = \phi(\Lambda^{-1} x)$$

This makes sense to me as "the transformed field at the transformed point in spacetime should be the same as the un-transformed field at the untransformed-transformed point in spacetime".

So trying to do that with inverse transformations, now using $\Lambda_{\frac{1}{2}}$ for a spinor plane-wave solution, I get

$$\Lambda u(p) = u (\Lambda^{-1} p)$$

and applying an inverse transformation would give

$$\Lambda^{-1} \Lambda u(p) = \Lambda^{-1} u (\Lambda^{-1} p)$$

or

$$u(\Lambda^{-1} \Lambda p) = u' (\Lambda^{-1} p)$$

so

$$u(\Lambda^{-1} p') = u( [\Lambda^{-1}]^{-1} \Lambda^{-1} p)$$

whence

$$u(\Lambda^{-1} p') = u(p)$$

that is,

$$u(p) = u(p)$$

So it's consistent alright, but not of much use!

Can anyone show me what I'm missing to derive equation (1) above. Thank you in advance!