Apparently you need to set the Projection for it to work correctly.
Here is an example of the code centering at London:
<html>
<head>
<script src="http://openweathermap.org/js/OWM.OpenLayers.1.3.4.js" ></script>
</head>
<body onload="init()">
<div id="basicMap"></div>
</body>
<script type="text/javascript">
function init() {
//Center of map
var lat = 51.5112139; // <--- some dynamic location
var lon = -0.1198244; //<--- some dynamic location
var map = new OpenLayers.Map("basicMap");
var fromProjection = new OpenLayers.Projection("EPSG:4326"); // Transform from WGS 1984
var toProjection = new OpenLayers.Projection("EPSG:900913"); // to Spherical Mercator Projection
var position = new OpenLayers.LonLat(lon, lat).transform( fromProjection, toProjection);
// Create overlays
// OSM
var mapnik = new OpenLayers.Layer.OSM();
// Stations
var stations = new OpenLayers.Layer.Vector.OWMStations("Stations");
// Current weather
var city = new OpenLayers.Layer.Vector.OWMWeather("Weather");
//Addind maps
map.addLayers([mapnik, stations, city]);
map.setCenter( position, 5 );
}
</script>
</html>
To get the Longitude and Latitude of your place, you can use this URL: http://www.mapcoordinates.net/en
Hope this helps