You should not depend your interface to any db context and initialize method. You can do it in concrete class constructor.

public interface IService
{
    void DoSomething();
    void DoOtherThing();
}

public class Service : IService
{
    private readonly object dependency1;
    private readonly object dependency2;
    private readonly object dependency3;
    private readonly object context;

    public Service(
        object dependency1,
        object dependency2,
        object dependency3,
        object context )
    {
        this.dependency1 = dependency1 ?? throw new ArgumentNullException(nameof(dependency1));
        this.dependency2 = dependency2 ?? throw new ArgumentNullException(nameof(dependency2));
        this.dependency3 = dependency3 ?? throw new ArgumentNullException(nameof(dependency3));
       
        // context is concrete class details not interfaces.
        this.context = context;
        
        // call init here constructor.
        this.Initialize(context);
    }

    protected void Initialize(Context context)
    {
        // Initialize state based on context
        // Heavy, long running operation
    }

    public void DoSomething()
    {
        // ...
    }

    public void DoOtherThing()
    {
        // ...
    }
}

By the way, if you use your Service class for long time without disposing, then you can call Initialize method on Static Constructor :

public class Service : IService
{
    private readonly object dependency1;
    private readonly object dependency2;
    private readonly object dependency3;
    private static object _context;

    public Service(
        object dependency1,
        object dependency2,
        object dependency3,
        object context )
    {
        this.dependency1 = dependency1 ?? throw new ArgumentNullException(nameof(dependency1));
        this.dependency2 = dependency2 ?? throw new ArgumentNullException(nameof(dependency2));
        this.dependency3 = dependency3 ?? throw new ArgumentNullException(nameof(dependency3));
        _context = context;

    }
    
    
    static Service()
    {
        Initialize(_context);
    }

    protected static void Initialize(Context context)
    {
        // Initialize state based on context
        // Heavy, long running operation
    }

    //...

}