When you need to construct a LineString object from C# you can use the SqlGeographyBuilder class located in the Microsoft.SqlServer.Types dll to help you do it. This is how:
var points = ...;//a list of points that you want to add;
var firstPoint = points[0];
var builder = new SqlGeographyBuilder();
builder.SetSrid(4326);
builder.BeginGeography(OpenGisGeographyType.LineString);
builder.BeginFigure(firstPoint.Lat, firstPoint.Lng);
points.GetRange(1, points.Count - 1).ForEach(point => builder.AddLine(point.Lat, point.Lng));
builder.EndFigure();
builder.EndGeography();
var lineString = builder.ConstructedGeography;