| L(s) = 1 | − 3-s + 4·5-s + 4·7-s − 2·9-s − 2·11-s − 4·15-s + 3·19-s − 4·21-s + 11·25-s + 5·27-s + 2·33-s + 16·35-s − 10·37-s − 5·41-s + 7·43-s − 8·45-s − 7·47-s + 9·49-s − 7·53-s − 8·55-s − 3·57-s − 5·59-s − 2·61-s − 8·63-s − 12·67-s + 3·71-s − 6·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.78·5-s + 1.51·7-s − 2/3·9-s − 0.603·11-s − 1.03·15-s + 0.688·19-s − 0.872·21-s + 11/5·25-s + 0.962·27-s + 0.348·33-s + 2.70·35-s − 1.64·37-s − 0.780·41-s + 1.06·43-s − 1.19·45-s − 1.02·47-s + 9/7·49-s − 0.961·53-s − 1.07·55-s − 0.397·57-s − 0.650·59-s − 0.256·61-s − 1.00·63-s − 1.46·67-s + 0.356·71-s − 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 305552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 305552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 113 | \( 1 - T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 + 7 T + p T^{2} \) | 1.47.h |
| 53 | \( 1 + 7 T + p T^{2} \) | 1.53.h |
| 59 | \( 1 + 5 T + p T^{2} \) | 1.59.f |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 13 T + p T^{2} \) | 1.79.an |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
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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
−12.91950616960659, −12.37207721211013, −11.94436579657537, −11.52847419056051, −10.83297505022084, −10.77673686760254, −10.27200549389267, −9.774877128513433, −9.137864841320945, −8.894124276105964, −8.293087733861065, −7.847912202315359, −7.322948973235299, −6.660511850150944, −6.210568440333636, −5.756706437503868, −5.290985373303852, −4.948173067541256, −4.765740706244828, −3.751523475468951, −2.970074733210251, −2.621370270970595, −1.802520809552836, −1.635624571912916, −0.9336147425640634, 0,
0.9336147425640634, 1.635624571912916, 1.802520809552836, 2.621370270970595, 2.970074733210251, 3.751523475468951, 4.765740706244828, 4.948173067541256, 5.290985373303852, 5.756706437503868, 6.210568440333636, 6.660511850150944, 7.322948973235299, 7.847912202315359, 8.293087733861065, 8.894124276105964, 9.137864841320945, 9.774877128513433, 10.27200549389267, 10.77673686760254, 10.83297505022084, 11.52847419056051, 11.94436579657537, 12.37207721211013, 12.91950616960659
