| L(s) = 1 | − 2-s + 4-s − 5-s − 8-s + 10-s − 6·13-s + 16-s + 2·17-s + 2·19-s − 20-s + 2·23-s − 4·25-s + 6·26-s + 6·29-s − 32-s − 2·34-s + 3·37-s − 2·38-s + 40-s − 9·41-s − 8·43-s − 2·46-s − 3·47-s + 4·50-s − 6·52-s + 4·53-s − 6·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s − 1.66·13-s + 1/4·16-s + 0.485·17-s + 0.458·19-s − 0.223·20-s + 0.417·23-s − 4/5·25-s + 1.17·26-s + 1.11·29-s − 0.176·32-s − 0.342·34-s + 0.493·37-s − 0.324·38-s + 0.158·40-s − 1.40·41-s − 1.21·43-s − 0.294·46-s − 0.437·47-s + 0.565·50-s − 0.832·52-s + 0.549·53-s − 0.787·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320166 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 11 T + p T^{2} \) | 1.79.al |
| 83 | \( 1 - T + p T^{2} \) | 1.83.ab |
| 89 | \( 1 + 5 T + p T^{2} \) | 1.89.f |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−12.56216762607462, −12.35114247439124, −11.82483968765224, −11.61481883862108, −11.03370833568303, −10.42353255976687, −10.06141735380944, −9.662323513440775, −9.350042095386663, −8.658177055053851, −8.173749210695169, −7.839464817879036, −7.398337647929744, −6.876315931463812, −6.558854908046533, −5.840619317672082, −5.277430510776565, −4.804183953949552, −4.419754787495928, −3.491836599869141, −3.220347897233601, −2.561714608596602, −2.013060263147023, −1.373724584998996, −0.6146476873289808, 0,
0.6146476873289808, 1.373724584998996, 2.013060263147023, 2.561714608596602, 3.220347897233601, 3.491836599869141, 4.419754787495928, 4.804183953949552, 5.277430510776565, 5.840619317672082, 6.558854908046533, 6.876315931463812, 7.398337647929744, 7.839464817879036, 8.173749210695169, 8.658177055053851, 9.350042095386663, 9.662323513440775, 10.06141735380944, 10.42353255976687, 11.03370833568303, 11.61481883862108, 11.82483968765224, 12.35114247439124, 12.56216762607462
