How do I request an oauth token in FitBit with WWW

I am trying to connect to FitBit using the WWW class in C#. When I try to get an oauth token, it fails with a "401 Unauthorized" message. I've tried directly connecting via curl with, as far as I can tell, the same headers and it's successful. Can anyone spot what I'm doing wrong?

When I enter this at the command line:

 curl -i -X 'POST' 'https://api.fitbit.com/oauth/request_token' -H 'Authorization: OAuth oauth_consumer_key="9e2fe319f5154a658731f73ce8373ef6", oauth_nonce="47F7793F", oauth_signature="u4Mvy%2BfR5rk8C%2BXERSz%2Bc%2F9ppj0%3D", oauth_signature_method="HMAC-SHA1", oauth_timestamp="1404279962", oauth_version="1.0"'

I Get:

 HTTP/1.1 200 OK
 Server: nginx
 Date: Wed, 02 Jul 2014 05:47:57 GMT
 Content-Type: application/x-www-form-urlencoded;charset=UTF-8
 Content-Length: 126
 Connection: keep-alive
 X-UA-Compatible: IE=edge,chrome=1
 Set-Cookie: fhttps=""; Expires=Thu, 01-Jan-1970 00:00:10 GMT; Path=/
 Expires: Thu, 01 Jan 1970 00:00:00 GMT
 Cache-control: no-store, no-cache, must-revalidate
 Pragma: no-cache
 Content-Language: en-US
 X-Frame-Options: SAMEORIGIN
 
 oauth_token=5c87cb5625ad6afdac957c7460d5c1d3&oauth_token_secret=73960a8e22f3c7c2cfb258fd8f83221e&oauth_callback_confirmed=true%                                                              

Using this code in Unity:

 using UnityEngine;
 using System;
 using System.Collections;
 using System.Text;
 using System.Text.RegularExpressions;
 
 public class Demo : MonoBehaviour 
 {
 
     public class RequestTokenResponse
     {
         public string Token { get; set; }
         public string TokenSecret { get; set; }
     }
 
     // You need to register your game or application in Fitbit to get cosumer key and secret.
     // Go to this page for registration: https://dev.fitbit.com/apps/new 
     public string CONSUMER_KEY = "9e2fe319f5154a658731f73ce8373ef6";
     public string CONSUMER_SECRET = "75f4e6d41d5f4469a49e8b929dd0b6b4";
     private string RequestTokenURL = "http://api.fitbit.com/oauth/request_token";
 
     // Use this for initialization
     IEnumerator Start() 
     {
         byte[] dummy = new byte[1];
         dummy[0] = 0;
 
         String timestamp = "1404279962";
         String signature = "u4Mvy+fR5rk8C+XERSz+c/9ppj0=";
 
         StringBuilder parameters = new StringBuilder();
         parameters.AppendFormat("OAuth oauth_consumer_key=\"{0}\",oauth_nonce=\"{1}\",oauth_signature=\"{2}\"," + 
                                        "oauth_signature_method=\"{3}\",oauth_timestamp=\"{4}\",oauth_version=\"{5}\"", 
                                        "9e2fe319f5154a658731f73ce8373ef6",
                                        "47F7793F",
                                         UrlEncode(signature),
                                        "HMAC-SHA1",
                                        timestamp,
                                        "1.0");
 
         Hashtable headers = new Hashtable();
         headers["Authorization"] = parameters.ToString();
         Debug.Log ("Authorization Header: " + headers["Authorization"]);
 
 
         WWW web = new WWW(RequestTokenURL, dummy, headers);
 
         yield return web;
 
         if (!string.IsNullOrEmpty(web.error))
         {
             Debug.Log(string.Format("GetRequestToken - failed. error : {0}", web.error));
         }
         else
         {
             String Token = Regex.Match(web.text, @"oauth_token=([^&]+)").Groups[1].Value;
             String TokenSecret = Regex.Match(web.text, @"oauth_token_secret=([^&]+)").Groups[1].Value;
             Debug.Log (string.Format ("Token: {0} TokenSecret: {1}", Token, TokenSecret));
         }
     }
         
     private static string UrlEncode(string value)
     {
         if (string.IsNullOrEmpty(value))
         {
             return string.Empty;
         }
             
         value = Uri.EscapeDataString(value);
             
         // UrlEncode escapes with lowercase characters (e.g. %2f) but oAuth needs %2F
         value = Regex.Replace(value, "(%[0-9a-f][0-9a-f])", c => c.Value.ToUpper());
             
         // these characters are not escaped by UrlEncode() but needed to be escaped
         value = value
             .Replace("(", "%28")
                 .Replace(")", "%29")
                 .Replace("$", "%24")
                 .Replace("!", "%21")
                 .Replace("*", "%2A")
                 .Replace("'", "%27");
         
         // these characters are escaped by UrlEncode() but will fail if unescaped!
         value = value.Replace("%7E", "~");
         
         return value;
     }
     
     // Update is called once per frame
     void Update() 
     {
     }
 }

When I execute it I get:

 Authorization Header: OAuth oauth_consumer_key="9e2fe319f5154a658731f73ce8373ef6",oauth_nonce="47F7793F",oauth_signature="u4Mvy%2BfR5rk8C%2BXERSz%2Bc%2F9ppj0%3D",oauth_signature_method="HMAC-SHA1",oauth_timestamp="1404279962",oauth_version="1.0"
 UnityEngine.Debug:Log(Object)
 <Start>c__Iterator0:MoveNext() (at Assets/Demo.cs:44)
 
 GetRequestToken - failed. error : 401 Unauthorized
 UnityEngine.Debug:Log(Object)
 <Start>c__Iterator0:MoveNext() (at Assets/Demo.cs:53)