I have a custom object on which their is a look to Account. Now i want to show all records in Service Provider Community where Custom field Account is equals to service provider's Account. I am not sure how can i do this using standard way the list view is standard list view and we can not create custom list view.
You'll want to use sharing sets. This will allow you to share all records that match the user's account with them. Note this will mean you need to make the public org wide sharing for that object private (the setting for external users, not internal). Also, sharing sets are only supported with Salesforce Community Licenses, if your org is like ours and you use a custom license (we have a really old org) these won't be supported.
Disclaimer: I'd normally quote the source material; however, it's kinda long. Also, if the material disappears that means its deprecated and no longer valid anyway.