| L(s) = 1 | − 2-s + 4-s + 5-s + 3·7-s − 8-s − 10-s + 4·11-s − 3·14-s + 16-s − 17-s − 2·19-s + 20-s − 4·22-s + 3·23-s + 25-s + 3·28-s − 8·29-s − 3·31-s − 32-s + 34-s + 3·35-s + 2·37-s + 2·38-s − 40-s − 8·41-s − 2·43-s + 4·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 1.13·7-s − 0.353·8-s − 0.316·10-s + 1.20·11-s − 0.801·14-s + 1/4·16-s − 0.242·17-s − 0.458·19-s + 0.223·20-s − 0.852·22-s + 0.625·23-s + 1/5·25-s + 0.566·28-s − 1.48·29-s − 0.538·31-s − 0.176·32-s + 0.171·34-s + 0.507·35-s + 0.328·37-s + 0.324·38-s − 0.158·40-s − 1.24·41-s − 0.304·43-s + 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 258570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 258570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 11 T + p T^{2} \) | 1.79.l |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−13.16218816334931, −12.52660903206819, −11.90508886301456, −11.58026287685972, −11.15109159357132, −10.87807049792176, −10.17796766017950, −9.819489937169956, −9.238596080540074, −8.896594317258076, −8.437908034061647, −8.082247435384228, −7.343104498421468, −7.006538571904160, −6.570999008433718, −5.987448106327452, −5.310650926980554, −5.129705760489731, −4.210583857876477, −3.913131363644352, −3.224386329324155, −2.417881595544591, −1.911463098439131, −1.490295944361233, −0.9262021742607878, 0,
0.9262021742607878, 1.490295944361233, 1.911463098439131, 2.417881595544591, 3.224386329324155, 3.913131363644352, 4.210583857876477, 5.129705760489731, 5.310650926980554, 5.987448106327452, 6.570999008433718, 7.006538571904160, 7.343104498421468, 8.082247435384228, 8.437908034061647, 8.896594317258076, 9.238596080540074, 9.819489937169956, 10.17796766017950, 10.87807049792176, 11.15109159357132, 11.58026287685972, 11.90508886301456, 12.52660903206819, 13.16218816334931
