In addition to Andreas's excellent answer, we should also mention the Tamagawa number conjecture of Bloch and Kato, which predicts the undetermined rational factor arising in Beilison's conjectural description of the $L$-value. The Bloch-Kato conjecture was later reformulated and generalized by Fontaine and Perrin-Riou to the case of motives with coefficients in an arbitrary number field. Here are some references :
Bloch, Kato, L-functions and Tamagawa numbers of motives.
Fontaine, Perrin-Riou, Autour des conjectures de Bloch et Kato: cohomologie galoisienne et valeurs de fonctions L.
Colmez, Fonctions L p-adiques.
Kings, The Bloch-Kato conjecture on special values of L-functions. A survey of known results.
Flach, The equivariant Tamagawa number conjecture : A survey.
Gealy, On the Tamagawa Number Conjecture for Motives Attached to Modular Forms.
Bellaïche, An introduction to the conjecture of Bloch and Kato.