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!