| L(s) = 1 | + 2-s + 4-s + 2·5-s + 7-s + 8-s + 2·10-s − 13-s + 14-s + 16-s − 4·19-s + 2·20-s + 4·23-s − 25-s − 26-s + 28-s + 4·31-s + 32-s + 2·35-s − 8·37-s − 4·38-s + 2·40-s + 6·41-s + 4·43-s + 4·46-s + 4·47-s + 49-s − 50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.377·7-s + 0.353·8-s + 0.632·10-s − 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.917·19-s + 0.447·20-s + 0.834·23-s − 1/5·25-s − 0.196·26-s + 0.188·28-s + 0.718·31-s + 0.176·32-s + 0.338·35-s − 1.31·37-s − 0.648·38-s + 0.316·40-s + 0.937·41-s + 0.609·43-s + 0.589·46-s + 0.583·47-s + 1/7·49-s − 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 198198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198198 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 16 T + p T^{2} \) | 1.89.q |
| 97 | \( 1 + 4 T + p T^{2} \) | 1.97.e |
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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
−13.39968468110604, −12.78654130460745, −12.45848200287268, −12.10262999564292, −11.38381692946253, −10.96162862280892, −10.62776108772170, −10.05763583317848, −9.611107919263662, −9.039844804985979, −8.636726091745899, −7.957488817312040, −7.550061768498281, −6.868330874189121, −6.520110057735808, −5.970482668230944, −5.470895942014856, −5.093355506289482, −4.401045598587332, −4.119096230461933, −3.282097675069367, −2.718899214312887, −2.232994492666126, −1.648613438570694, −1.054660833822801, 0,
1.054660833822801, 1.648613438570694, 2.232994492666126, 2.718899214312887, 3.282097675069367, 4.119096230461933, 4.401045598587332, 5.093355506289482, 5.470895942014856, 5.970482668230944, 6.520110057735808, 6.868330874189121, 7.550061768498281, 7.957488817312040, 8.636726091745899, 9.039844804985979, 9.611107919263662, 10.05763583317848, 10.62776108772170, 10.96162862280892, 11.38381692946253, 12.10262999564292, 12.45848200287268, 12.78654130460745, 13.39968468110604
