| L(s) = 1 | − 2·3-s − 2·7-s + 9-s − 2·11-s + 2·13-s − 4·19-s + 4·21-s + 2·23-s − 5·25-s + 4·27-s − 6·29-s + 2·31-s + 4·33-s − 6·37-s − 4·39-s + 6·41-s − 4·43-s − 8·47-s − 3·49-s − 6·53-s + 8·57-s − 4·59-s + 2·61-s − 2·63-s + 12·67-s − 4·69-s + 12·71-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.755·7-s + 1/3·9-s − 0.603·11-s + 0.554·13-s − 0.917·19-s + 0.872·21-s + 0.417·23-s − 25-s + 0.769·27-s − 1.11·29-s + 0.359·31-s + 0.696·33-s − 0.986·37-s − 0.640·39-s + 0.937·41-s − 0.609·43-s − 1.16·47-s − 3/7·49-s − 0.824·53-s + 1.05·57-s − 0.520·59-s + 0.256·61-s − 0.251·63-s + 1.46·67-s − 0.481·69-s + 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 383792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 383792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 17 | \( 1 \) | |
| 83 | \( 1 - T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 16 T + p T^{2} \) | 1.73.aq |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 + p T^{2} \) | 1.97.a |
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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
−12.66627589949638, −12.35012640354506, −11.67510687138437, −11.28049965918154, −10.92383352416487, −10.66325627600551, −9.942345338268593, −9.699050362050347, −9.189232194378617, −8.555906817819939, −8.077247541886910, −7.754281123047530, −6.903953067103787, −6.595579994768081, −6.296114954482879, −5.716294669611592, −5.304778891830409, −4.916780647928017, −4.271029479173344, −3.626955392504609, −3.329106273216719, −2.524212666635435, −2.003086029922655, −1.292863021375565, −0.4796905449311351, 0,
0.4796905449311351, 1.292863021375565, 2.003086029922655, 2.524212666635435, 3.329106273216719, 3.626955392504609, 4.271029479173344, 4.916780647928017, 5.304778891830409, 5.716294669611592, 6.296114954482879, 6.595579994768081, 6.903953067103787, 7.754281123047530, 8.077247541886910, 8.555906817819939, 9.189232194378617, 9.699050362050347, 9.942345338268593, 10.66325627600551, 10.92383352416487, 11.28049965918154, 11.67510687138437, 12.35012640354506, 12.66627589949638
