| L(s) = 1 | − 3·3-s + 5-s − 5·7-s + 6·9-s + 13-s − 3·15-s − 3·17-s + 15·21-s + 2·23-s − 4·25-s − 9·27-s + 8·31-s − 5·35-s − 11·37-s − 3·39-s − 6·41-s + 11·43-s + 6·45-s + 7·47-s + 18·49-s + 9·51-s + 14·53-s − 6·59-s + 10·61-s − 30·63-s + 65-s − 8·67-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.447·5-s − 1.88·7-s + 2·9-s + 0.277·13-s − 0.774·15-s − 0.727·17-s + 3.27·21-s + 0.417·23-s − 4/5·25-s − 1.73·27-s + 1.43·31-s − 0.845·35-s − 1.80·37-s − 0.480·39-s − 0.937·41-s + 1.67·43-s + 0.894·45-s + 1.02·47-s + 18/7·49-s + 1.26·51-s + 1.92·53-s − 0.781·59-s + 1.28·61-s − 3.77·63-s + 0.124·65-s − 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| 19 | \( 1 \) | |
| good | 3 | \( 1 + p T + p T^{2} \) | 1.3.d |
| 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 + 5 T + p T^{2} \) | 1.7.f |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 11 T + p T^{2} \) | 1.37.l |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 11 T + p T^{2} \) | 1.43.al |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 7 T + p T^{2} \) | 1.71.h |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−12.87276956392267, −12.32040137865547, −11.93430338719462, −11.81634082265454, −10.95343769809168, −10.56199577208745, −10.28686515380213, −9.921299040545028, −9.323506325077509, −8.960111818455389, −8.437243666471817, −7.378809494934596, −7.223673092641164, −6.566675694237317, −6.331899565916827, −5.956072855394052, −5.477690576540805, −5.017972468531489, −4.331377565970190, −3.862106827829011, −3.346070911238167, −2.568628743656724, −2.071137899453535, −1.118163892146624, −0.6128914476843344, 0,
0.6128914476843344, 1.118163892146624, 2.071137899453535, 2.568628743656724, 3.346070911238167, 3.862106827829011, 4.331377565970190, 5.017972468531489, 5.477690576540805, 5.956072855394052, 6.331899565916827, 6.566675694237317, 7.223673092641164, 7.378809494934596, 8.437243666471817, 8.960111818455389, 9.323506325077509, 9.921299040545028, 10.28686515380213, 10.56199577208745, 10.95343769809168, 11.81634082265454, 11.93430338719462, 12.32040137865547, 12.87276956392267
