you could replace the sum by table to see the individual terms for each combination of j, jp, lambdaW, lambdaWp. Try
lh = Flatten[
Table[{{j,
jp, \[Lambda]w, \[Lambda]wp}, (-1)^(j + jp) h1[ \[Lambda]1,
1/2] KroneckerDelta[\[Lambda]2 - \[Lambda]w, \[Lambda]2 - \
\[Lambda]wp] d3[j, \[Lambda]w, \[Lambda]1 - (1/2)] d3[
jp, \[Lambda]wp, \[Lambda]1 - (1/
2)] h[\[Lambda]2, \[Lambda]w] Conjugate[
h[\[Lambda]2, \[Lambda]wp]]}, {j, 0, 1}, {jp, 0,
1}, {\[Lambda]w, {1, 0, -1}}, {\[Lambda]wp, {1,
0, -1}}, {\[Lambda]1, {-1/2, 1/2}}, {\[Lambda]2, {-1/2, 1/2}}],
5]
To look at the result you could use
ableForm[lh, TableHeadings -> {None, {index, f}}, TableDepth -> 2]
You can always latter sum up the term from the list. Does this help your analysis?