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.

Bellaïche, An introduction to the conjecture of Bloch and Kato.

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.

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 arbitrary coefficients. 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.

Flach, The equivariant Tamagawa number conjecture : A survey.

Bellaïche, An introduction to the conjecture of Bloch and Kato.

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.

Bellaïche, An introduction to the conjecture of Bloch and Kato.