| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s + 9-s − 12-s + 13-s + 14-s + 16-s − 4·17-s − 18-s + 2·19-s + 21-s − 4·23-s + 24-s − 5·25-s − 26-s − 27-s − 28-s − 2·29-s − 10·31-s − 32-s + 4·34-s + 36-s − 10·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.288·12-s + 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.970·17-s − 0.235·18-s + 0.458·19-s + 0.218·21-s − 0.834·23-s + 0.204·24-s − 25-s − 0.196·26-s − 0.192·27-s − 0.188·28-s − 0.371·29-s − 1.79·31-s − 0.176·32-s + 0.685·34-s + 1/6·36-s − 1.64·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 66066 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66066 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−14.52660409702385, −13.88351638415681, −13.40899625358569, −12.85894248008358, −12.39204244289389, −11.74788364882608, −11.44201870130881, −10.90154964091471, −10.30775866559691, −10.02290645090393, −9.284276212177929, −8.937748076022744, −8.398808744997351, −7.623090240442228, −7.245712054737719, −6.731307347734679, −6.119503830815389, −5.575502251387309, −5.190037924042571, −4.103952410069237, −3.842018160186712, −3.037506622310071, −2.052637302237540, −1.790377371082907, −0.6782215454640609, 0,
0.6782215454640609, 1.790377371082907, 2.052637302237540, 3.037506622310071, 3.842018160186712, 4.103952410069237, 5.190037924042571, 5.575502251387309, 6.119503830815389, 6.731307347734679, 7.245712054737719, 7.623090240442228, 8.398808744997351, 8.937748076022744, 9.284276212177929, 10.02290645090393, 10.30775866559691, 10.90154964091471, 11.44201870130881, 11.74788364882608, 12.39204244289389, 12.85894248008358, 13.40899625358569, 13.88351638415681, 14.52660409702385
