No. Your landlord probably has gotten a homeowner insurance to protect their property. Therefore, your laptop is not under the coverage of the homeowner insurance.
But you, as a renter, will need a renters insurance to cover your laptop and other personal property.
The information in this answer is general in nature and is not legal or insurance advice for your specific situation. Policy language varies carrier by carrier; your declarations page is the final word. If you're unsure, call your agent before you need to file.