| L(s) = 1 | − 3·3-s − 5-s − 7-s + 6·9-s + 11-s + 3·15-s − 3·17-s + 19-s + 3·21-s + 2·23-s + 25-s − 9·27-s + 29-s − 3·31-s − 3·33-s + 35-s − 7·37-s + 10·41-s + 8·43-s − 6·45-s + 2·47-s − 6·49-s + 9·51-s − 53-s − 55-s − 3·57-s + 6·59-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.447·5-s − 0.377·7-s + 2·9-s + 0.301·11-s + 0.774·15-s − 0.727·17-s + 0.229·19-s + 0.654·21-s + 0.417·23-s + 1/5·25-s − 1.73·27-s + 0.185·29-s − 0.538·31-s − 0.522·33-s + 0.169·35-s − 1.15·37-s + 1.56·41-s + 1.21·43-s − 0.894·45-s + 0.291·47-s − 6/7·49-s + 1.26·51-s − 0.137·53-s − 0.134·55-s − 0.397·57-s + 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 148720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 148720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 + p T + p T^{2} \) | 1.3.d |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 3 T + p T^{2} \) | 1.61.ad |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + T + p T^{2} \) | 1.71.b |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 10 T + p T^{2} \) | 1.83.ak |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 97 | \( 1 + 4 T + p T^{2} \) | 1.97.e |
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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
−13.40940548188903, −12.80796847261486, −12.62536432605552, −12.11712005647096, −11.63158772226944, −11.18347270817162, −10.90013555241680, −10.41378486175681, −9.901683567734063, −9.247248758770443, −8.962335672792404, −8.159805685583992, −7.531943316035698, −7.079405436549171, −6.669274466078430, −6.151162572788888, −5.759755745795533, −4.992885161709636, −4.870623751956110, −3.929801076087622, −3.822947494757995, −2.820883446370082, −2.109688538027411, −1.240988948519838, −0.6840331095534201, 0,
0.6840331095534201, 1.240988948519838, 2.109688538027411, 2.820883446370082, 3.822947494757995, 3.929801076087622, 4.870623751956110, 4.992885161709636, 5.759755745795533, 6.151162572788888, 6.669274466078430, 7.079405436549171, 7.531943316035698, 8.159805685583992, 8.962335672792404, 9.247248758770443, 9.901683567734063, 10.41378486175681, 10.90013555241680, 11.18347270817162, 11.63158772226944, 12.11712005647096, 12.62536432605552, 12.80796847261486, 13.40940548188903
