lambda_error_response_function.js

var errorResponse = function (statusCode, errorCode, detailMessage) {
    var messageMap = {
        400: 'Bad Request',
        401: 'Unauthonized',
        403: 'Forbidden',
        404: 'Not Found',
        405: 'Method Not Allowd',
        406: 'Not Acceptable',
        409: 'Conflict',
        500: 'Internal Server Error'
    };

    return {
        statusCode: statusCode,
        body: JSON.stringify({
            code: errorCode,
            message: messageMap[statusCode],
            detail: detailMessage
        }),
        headers: {
            'Content-Type': 'application/json',
            'X-Requested-With' : 'XMLHttpRequest',
            'Access-Control-Allow-Origin' : '*'
        }
    }
};

This is part of my lambda function code, Which allow return HTTP status code with message / enable CORS.
Of course you must check 'lambda proxy integration' in API-Gateway.