| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s + 9-s + 3·11-s − 12-s + 14-s + 16-s + 4·17-s − 18-s + 7·19-s + 21-s − 3·22-s + 2·23-s + 24-s − 27-s − 28-s − 2·29-s − 5·31-s − 32-s − 3·33-s − 4·34-s + 36-s − 4·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.904·11-s − 0.288·12-s + 0.267·14-s + 1/4·16-s + 0.970·17-s − 0.235·18-s + 1.60·19-s + 0.218·21-s − 0.639·22-s + 0.417·23-s + 0.204·24-s − 0.192·27-s − 0.188·28-s − 0.371·29-s − 0.898·31-s − 0.176·32-s − 0.522·33-s − 0.685·34-s + 1/6·36-s − 0.657·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 3 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 2 T + p T^{2} \) | 1.59.c |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + T + p T^{2} \) | 1.67.b |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 11 T + p T^{2} \) | 1.89.al |
| 97 | \( 1 - T + p T^{2} \) | 1.97.ab |
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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
−13.45084398281506, −12.80204043786574, −12.19266948297150, −12.00853565067624, −11.52523102642505, −11.08729199112374, −10.50686954440176, −10.01819068260138, −9.654809489566747, −9.217524514418409, −8.769475717075567, −8.097463193272560, −7.609353416518023, −7.054797894772949, −6.800508811837863, −6.171208340124837, −5.560878721244815, −5.264563060807470, −4.601810204432939, −3.745718812657262, −3.329020462374236, −2.934465255393909, −1.749329066915663, −1.509275279121755, −0.7740339446070341, 0,
0.7740339446070341, 1.509275279121755, 1.749329066915663, 2.934465255393909, 3.329020462374236, 3.745718812657262, 4.601810204432939, 5.264563060807470, 5.560878721244815, 6.171208340124837, 6.800508811837863, 7.054797894772949, 7.609353416518023, 8.097463193272560, 8.769475717075567, 9.217524514418409, 9.654809489566747, 10.01819068260138, 10.50686954440176, 11.08729199112374, 11.52523102642505, 12.00853565067624, 12.19266948297150, 12.80204043786574, 13.45084398281506
