| L(s) = 1 | + 2-s + 2·3-s + 4-s + 2·6-s + 4·7-s + 8-s + 9-s − 6·11-s + 2·12-s − 13-s + 4·14-s + 16-s + 6·17-s + 18-s + 2·19-s + 8·21-s − 6·22-s − 6·23-s + 2·24-s − 26-s − 4·27-s + 4·28-s − 6·29-s + 2·31-s + 32-s − 12·33-s + 6·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.816·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s − 1.80·11-s + 0.577·12-s − 0.277·13-s + 1.06·14-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.458·19-s + 1.74·21-s − 1.27·22-s − 1.25·23-s + 0.408·24-s − 0.196·26-s − 0.769·27-s + 0.755·28-s − 1.11·29-s + 0.359·31-s + 0.176·32-s − 2.08·33-s + 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.384717314\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.384717314\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−10.54706085367247596930787344294, −9.795340002438148581315476347893, −8.459740217786738812033872338622, −7.84813165184443739324894783244, −7.48067692981350872082789436709, −5.60849552610161287200501685502, −5.12551463257983439866136237124, −3.86701613615803161105270189660, −2.78612768174608789776745831075, −1.87624790774678570500324840839,
1.87624790774678570500324840839, 2.78612768174608789776745831075, 3.86701613615803161105270189660, 5.12551463257983439866136237124, 5.60849552610161287200501685502, 7.48067692981350872082789436709, 7.84813165184443739324894783244, 8.459740217786738812033872338622, 9.795340002438148581315476347893, 10.54706085367247596930787344294
