L(s) = 1 | − 2-s − 5·3-s + 5·6-s + 7-s + 15·9-s − 5·11-s + 5·13-s − 14-s − 17-s − 15·18-s − 5·21-s + 5·22-s + 2·23-s − 25-s − 5·26-s − 35·27-s + 2·31-s + 25·33-s + 34-s − 25·39-s + 5·42-s − 2·46-s + 50-s + 5·51-s − 2·53-s + 35·54-s − 2·62-s + ⋯ |
L(s) = 1 | − 2-s − 5·3-s + 5·6-s + 7-s + 15·9-s − 5·11-s + 5·13-s − 14-s − 17-s − 15·18-s − 5·21-s + 5·22-s + 2·23-s − 25-s − 5·26-s − 35·27-s + 2·31-s + 25·33-s + 34-s − 25·39-s + 5·42-s − 2·46-s + 50-s + 5·51-s − 2·53-s + 35·54-s − 2·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 7^{12} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 7^{12} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.0003348519134\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0003348519134\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 7 | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 17 | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
good | 3 | \( ( 1 + T )^{6}( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} ) \) |
| 5 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \) |
| 11 | \( ( 1 + T )^{6}( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} ) \) |
| 13 | \( ( 1 - T )^{6}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \) |
| 19 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
| 23 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \) |
| 29 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \) |
| 31 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \) |
| 37 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \) |
| 41 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \) |
| 43 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \) |
| 47 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \) |
| 53 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \) |
| 59 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \) |
| 61 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \) |
| 67 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
| 71 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \) |
| 73 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \) |
| 79 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \) |
| 83 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \) |
| 89 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \) |
| 97 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.92891450562438377191137973868, −4.54704064774992225096683047210, −4.40656388852732673871962817782, −4.37836332818426275165427131530, −4.37667816082238214350833080187, −4.01952632883530179143597315265, −4.01935910638189237534916565990, −3.70971037076037997130686393964, −3.55658383884694884193878397880, −3.31969567680309449118775789301, −3.29066087686150034731770777356, −3.03205587319924430838075138902, −2.95024773083053095708513365951, −2.75537476225048687331908415653, −2.11087475669093453480585777922, −2.10306961756013804235488997560, −2.05500255540446985087323566083, −1.98518704187982317773688742129, −1.79600152205025435696697965688, −1.17423292886629159964779486587, −1.15505477340255164107469844123, −1.06748305101418593736976033204, −0.936295276039083458896097147773, −0.74815926097795494371333911863, −0.01584294294396166476534273163,
0.01584294294396166476534273163, 0.74815926097795494371333911863, 0.936295276039083458896097147773, 1.06748305101418593736976033204, 1.15505477340255164107469844123, 1.17423292886629159964779486587, 1.79600152205025435696697965688, 1.98518704187982317773688742129, 2.05500255540446985087323566083, 2.10306961756013804235488997560, 2.11087475669093453480585777922, 2.75537476225048687331908415653, 2.95024773083053095708513365951, 3.03205587319924430838075138902, 3.29066087686150034731770777356, 3.31969567680309449118775789301, 3.55658383884694884193878397880, 3.70971037076037997130686393964, 4.01935910638189237534916565990, 4.01952632883530179143597315265, 4.37667816082238214350833080187, 4.37836332818426275165427131530, 4.40656388852732673871962817782, 4.54704064774992225096683047210, 4.92891450562438377191137973868
Plot not available for L-functions of degree greater than 10.