| L(s) = 1 | + 3-s − 5-s − 7-s + 9-s − 11-s − 2·13-s − 15-s + 6·17-s − 4·19-s − 21-s + 25-s + 27-s − 6·29-s + 4·31-s − 33-s + 35-s − 2·37-s − 2·39-s + 6·41-s − 4·43-s − 45-s + 49-s + 6·51-s − 6·53-s + 55-s − 4·57-s − 2·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.554·13-s − 0.258·15-s + 1.45·17-s − 0.917·19-s − 0.218·21-s + 1/5·25-s + 0.192·27-s − 1.11·29-s + 0.718·31-s − 0.174·33-s + 0.169·35-s − 0.328·37-s − 0.320·39-s + 0.937·41-s − 0.609·43-s − 0.149·45-s + 1/7·49-s + 0.840·51-s − 0.824·53-s + 0.134·55-s − 0.529·57-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| good | 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−14.39392007764324, −13.88966870852961, −13.27189397246388, −12.85878381276563, −12.32017527593266, −12.04273173908897, −11.29796207822139, −10.79636233405914, −10.16373223577404, −9.859530670332846, −9.256030762780670, −8.741727004508118, −8.142437313741597, −7.653874595544086, −7.361482768713004, −6.617748963475018, −6.052521527577739, −5.444777459567787, −4.769087553088874, −4.253322524651074, −3.493508301991097, −3.178591579345979, −2.423765750699929, −1.791618581313275, −0.8791561332701589, 0,
0.8791561332701589, 1.791618581313275, 2.423765750699929, 3.178591579345979, 3.493508301991097, 4.253322524651074, 4.769087553088874, 5.444777459567787, 6.052521527577739, 6.617748963475018, 7.361482768713004, 7.653874595544086, 8.142437313741597, 8.741727004508118, 9.256030762780670, 9.859530670332846, 10.16373223577404, 10.79636233405914, 11.29796207822139, 12.04273173908897, 12.32017527593266, 12.85878381276563, 13.27189397246388, 13.88966870852961, 14.39392007764324
