| L(s) = 1 | + 2-s − 4-s + 2·5-s − 3·8-s + 2·10-s + 6·11-s − 6·13-s − 16-s + 2·17-s − 19-s − 2·20-s + 6·22-s − 6·23-s − 25-s − 6·26-s + 6·29-s − 8·31-s + 5·32-s + 2·34-s − 2·37-s − 38-s − 6·40-s − 8·43-s − 6·44-s − 6·46-s + 8·47-s − 50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.894·5-s − 1.06·8-s + 0.632·10-s + 1.80·11-s − 1.66·13-s − 1/4·16-s + 0.485·17-s − 0.229·19-s − 0.447·20-s + 1.27·22-s − 1.25·23-s − 1/5·25-s − 1.17·26-s + 1.11·29-s − 1.43·31-s + 0.883·32-s + 0.342·34-s − 0.328·37-s − 0.162·38-s − 0.948·40-s − 1.21·43-s − 0.904·44-s − 0.884·46-s + 1.16·47-s − 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8379 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8379 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) | |
| 7 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 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
−7.23345997293065820415198566041, −6.62581522472163352001088769627, −5.86457100704306789503027080761, −5.45721547895839729519316087515, −4.54154552955351369529142238877, −4.06482317085525828355758382154, −3.22299135002408101319055649432, −2.27958244762200813940683648068, −1.42463212923715253077410185699, 0,
1.42463212923715253077410185699, 2.27958244762200813940683648068, 3.22299135002408101319055649432, 4.06482317085525828355758382154, 4.54154552955351369529142238877, 5.45721547895839729519316087515, 5.86457100704306789503027080761, 6.62581522472163352001088769627, 7.23345997293065820415198566041
