| L(s) = 1 | − 2-s + 2·3-s + 4-s − 5-s − 2·6-s − 8-s + 9-s + 10-s − 6·11-s + 2·12-s − 13-s − 2·15-s + 16-s + 6·17-s − 18-s − 2·19-s − 20-s + 6·22-s + 6·23-s − 2·24-s + 25-s + 26-s − 4·27-s − 6·29-s + 2·30-s − 2·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.447·5-s − 0.816·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.80·11-s + 0.577·12-s − 0.277·13-s − 0.516·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s − 0.458·19-s − 0.223·20-s + 1.27·22-s + 1.25·23-s − 0.408·24-s + 1/5·25-s + 0.196·26-s − 0.769·27-s − 1.11·29-s + 0.365·30-s − 0.359·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.511885857\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.511885857\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−7.84794214530463418836065833108, −7.68860379182053172287428510692, −7.13939958410469984199748452423, −5.80736345854100722521276598034, −5.32902805530358142021478868252, −4.22230316966172287572578732771, −3.23388672350047301815023131459, −2.78338279291491207419517143178, −2.00162834001726516894330272406, −0.64394251803466591925986785147,
0.64394251803466591925986785147, 2.00162834001726516894330272406, 2.78338279291491207419517143178, 3.23388672350047301815023131459, 4.22230316966172287572578732771, 5.32902805530358142021478868252, 5.80736345854100722521276598034, 7.13939958410469984199748452423, 7.68860379182053172287428510692, 7.84794214530463418836065833108
