contactListLocalSort.html

<template>
  <lightning-datatable 
  key-field="Id" 
  data={results} 
  columns={columns} 
  hide-checkbox-column	="false"
  show-row-number-column="true"
  onsort={updateColumnSorting}
  sorted-by={sortBy}
  sorted-direction={sortDirection}
  row-number-offset	="0" >
 </lightning-datatable>
</template>

contactListLocalSort.js

import { LightningElement,track,wire } from 'lwc';
import fetchContact from '@salesforce/apex/ContactList.fetchContactLocal';
const dataTablecolumns = [{
  label: 'First Name', 
  fieldName: 'FirstName',
  sortable : true, 
  type: 'text'
},
{
  label : 'Last Name',
  fieldName : 'LastName',
  type : 'text',
  sortable : true
}]
export default class ContactListLocalSort extends LightningElement {
  @track results=[];
  @track columns = dataTablecolumns;
  @track sortBy='FirstName';
  @track sortDirection='asc';
  @wire(fetchContact) contactList({error, data}) {
    if(data)
      this.results=Object.assign([], data);
    if(error)
      console.log(error);
  }
  updateColumnSorting(event){
    let fieldName = event.detail.fieldName;
    let sortDirection = event.detail.sortDirection;
    //assign the values
    this.sortBy = fieldName;
    this.sortDirection = sortDirection;
    //call the custom sort method.
    this.sortData(fieldName, sortDirection);
  }
  
  //This sorting logic here is very simple. This will be useful only for text or number field.
  // You will need to implement custom logic for handling different types of field.
  sortData(fieldName, sortDirection) {
    let sortResult = Object.assign([], this.results);
    this.results = sortResult.sort(function(a,b){
      if(a[fieldName] < b[fieldName])
        return sortDirection === 'asc' ? -1 : 1;
      else if(a[fieldName] > b[fieldName])
        return sortDirection === 'asc' ? 1 : -1;
      else
        return 0;
    })
  }
}