| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s − 2·11-s − 12-s + 13-s + 16-s − 18-s − 19-s + 2·22-s − 2·23-s + 24-s − 26-s − 27-s − 5·29-s + 6·31-s − 32-s + 2·33-s + 36-s − 5·37-s + 38-s − 39-s + 3·41-s + 4·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.603·11-s − 0.288·12-s + 0.277·13-s + 1/4·16-s − 0.235·18-s − 0.229·19-s + 0.426·22-s − 0.417·23-s + 0.204·24-s − 0.196·26-s − 0.192·27-s − 0.928·29-s + 1.07·31-s − 0.176·32-s + 0.348·33-s + 1/6·36-s − 0.821·37-s + 0.162·38-s − 0.160·39-s + 0.468·41-s + 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 11 T + p T^{2} \) | 1.53.al |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 12 T + p T^{2} \) | 1.61.am |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−13.99898392151280, −13.47282320447204, −13.05944797150391, −12.32553244789273, −12.16711281730632, −11.45978250690891, −10.93239989618416, −10.75252426897683, −9.997970847492199, −9.732246426204172, −9.156200212625300, −8.347583698888861, −8.233928361114988, −7.531470597502105, −6.911268723843904, −6.615169575185712, −5.836998232005783, −5.473362884504332, −4.921873159466571, −4.047111801856158, −3.708039571515218, −2.705013661699055, −2.285653830184039, −1.470218998365028, −0.7606942078051678, 0,
0.7606942078051678, 1.470218998365028, 2.285653830184039, 2.705013661699055, 3.708039571515218, 4.047111801856158, 4.921873159466571, 5.473362884504332, 5.836998232005783, 6.615169575185712, 6.911268723843904, 7.531470597502105, 8.233928361114988, 8.347583698888861, 9.156200212625300, 9.732246426204172, 9.997970847492199, 10.75252426897683, 10.93239989618416, 11.45978250690891, 12.16711281730632, 12.32553244789273, 13.05944797150391, 13.47282320447204, 13.99898392151280
