Delicate fabrics reward a little care. Wash lawn on a cool, gentle cycle and press it while slightly damp for a crisp finish. Silk is happiest hand washed in cool water with a mild detergent, then dried flat away from direct sun.
Always test a small offcut first, especially with bold colours, and prewash before you sew so any shrinkage happens before the garment is made.
Treat the fabric well and a good print will last for years.