RSA: Is there a way to digitally sign a message without knowing the private key?
The comment by @Krystian is spot on. Textbook RSA is multiplicative so since $m_3=m_1 m_2,$ then $s_3=m_1^d m_2^d \pmod n=s_1 s_2 \pmod n.$
Magma tells me that $s_3$ should be

5483355855153602627619220309939701891434212768777923480007006960005208299649205
6581008264568421280116806103922182738655624242961539966642663712352455540402732
9086686980626004891264773131893337185873828146733471102601951184587947342940793
37057046673879975457092965542603806917235403924230185514038638412396010

if I haven't typed incorrectly.