| L(s) = 1 | − 2·4-s + 5-s + 3·11-s + 4·16-s + 3·17-s − 4·19-s − 2·20-s − 9·23-s + 25-s − 6·29-s + 2·31-s + 37-s − 3·41-s + 2·43-s − 6·44-s − 6·47-s + 9·53-s + 3·55-s − 12·59-s − 5·61-s − 8·64-s + 4·67-s − 6·68-s − 9·71-s + 14·73-s + 8·76-s − 7·79-s + ⋯ |
| L(s) = 1 | − 4-s + 0.447·5-s + 0.904·11-s + 16-s + 0.727·17-s − 0.917·19-s − 0.447·20-s − 1.87·23-s + 1/5·25-s − 1.11·29-s + 0.359·31-s + 0.164·37-s − 0.468·41-s + 0.304·43-s − 0.904·44-s − 0.875·47-s + 1.23·53-s + 0.404·55-s − 1.56·59-s − 0.640·61-s − 64-s + 0.488·67-s − 0.727·68-s − 1.06·71-s + 1.63·73-s + 0.917·76-s − 0.787·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 372645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 372645 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 3 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 9 T + p T^{2} \) | 1.23.j |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 7 T + p T^{2} \) | 1.79.h |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 - 5 T + p T^{2} \) | 1.97.af |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.68756688538012, −12.28317752764143, −11.95444583414218, −11.45916637866964, −10.76751549637130, −10.38949290512667, −9.901272215656874, −9.550047957820448, −9.197400647693693, −8.565098778141883, −8.346956843225222, −7.681267843831995, −7.384437389947636, −6.531133435811811, −6.137215883576181, −5.844545626727349, −5.238147766461721, −4.676343579287301, −4.235964565541810, −3.697005213163633, −3.419411352207446, −2.548355489584348, −1.870478669320691, −1.477218004289689, −0.6789478625024501, 0,
0.6789478625024501, 1.477218004289689, 1.870478669320691, 2.548355489584348, 3.419411352207446, 3.697005213163633, 4.235964565541810, 4.676343579287301, 5.238147766461721, 5.844545626727349, 6.137215883576181, 6.531133435811811, 7.384437389947636, 7.681267843831995, 8.346956843225222, 8.565098778141883, 9.197400647693693, 9.550047957820448, 9.901272215656874, 10.38949290512667, 10.76751549637130, 11.45916637866964, 11.95444583414218, 12.28317752764143, 12.68756688538012
