expository articles on special values of L functions While searching for some notes on L functions i have seen the following statement...
In mathematics, the study of special values of L-functions is a sub field of number theory devoted to generalizing formulae such as the Leibniz formula for pi, namely
$$1 \,-\, \frac{1}{3} \,+\, \frac{1}{5} \,-\, \frac{1}{7} \,+\, \frac{1}{9} \,-\, \cdots \;=\; \frac{\pi}{4}$$
by the recognition that expression on the left-hand side is also $L(1)$ where $L(s)$ is the Dirichlet L-function for the Gaussian field.
Please suggest some basic references and links of some expository articles on these special values of L functions..
 A: The modern theory is both very deep and very broad, so there won't be a truly elementary introduction that gives you a feel for all of its dimensions.  For example, two huge conjectures about special values of L-functions are
1) The Conjecture of Birch and Swinnerton-Dyer
2) The Stark conjectures
(The first, being a millenium prize problem, is discussed by Keith Devlin in his book "The Millenium Prize Problems")  Getting a solid understanding of even these topics requires a very large amount of background.  And both of these are subsumed by the equivariant Tamagawa number conjecture, which is understood in full generality only by a few specialists.
Fairly expository articles on all of these conjectures can be found in the Park City lecture notes on the topic: "Arithmetic of L-functions", edited by Popescu, Rubin, and Silverberg, Volume 18 in the IAS/Park City mathematics series. Expository here means that the audience authors were supposed to have in mind is graduate students.  
Kolster gives a lucid introduction to special values of zeta functions in particular in the introduction to
http://users.ictp.it/~pub_off/lectures/lns015/Kolster/Kolster_Final.pdf
If you are willing to to a lot of mental filing of "okay, there's some mathematical object called a 'thingy'", that introduction can give you a quick feel for the modern depth of the subject.
If you are willing to settle for learning classical knowledge of special values of L-functions, there are lots of good sources. A particularly nice, elementary discussion of Euler's original computation of zeta(2) is in William Dunham's excellent book, "Journey Through Genius".  
