| L(s) = 1 | + 2-s + 4-s − 5-s + 7-s + 8-s − 10-s + 14-s + 16-s + 2·17-s − 20-s + 6·23-s + 25-s + 28-s − 6·29-s + 2·31-s + 32-s + 2·34-s − 35-s + 2·37-s − 40-s + 10·41-s − 12·43-s + 6·46-s − 4·47-s + 49-s + 50-s − 12·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s + 0.353·8-s − 0.316·10-s + 0.267·14-s + 1/4·16-s + 0.485·17-s − 0.223·20-s + 1.25·23-s + 1/5·25-s + 0.188·28-s − 1.11·29-s + 0.359·31-s + 0.176·32-s + 0.342·34-s − 0.169·35-s + 0.328·37-s − 0.158·40-s + 1.56·41-s − 1.82·43-s + 0.884·46-s − 0.583·47-s + 1/7·49-s + 0.141·50-s − 1.64·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 106470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 106470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 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.ac |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−14.08572393650125, −13.23980352982162, −12.99093937289402, −12.65695062139560, −11.88658741265851, −11.59299511860467, −11.01181703901586, −10.86972167807858, −9.915285047549310, −9.682320050318862, −8.930921114032481, −8.352918390722232, −7.934219308541970, −7.339663930455842, −6.939619144886551, −6.349073337121515, −5.699123116179744, −5.215779702216174, −4.722988536643146, −4.167699998049387, −3.563091082902004, −3.041936789854456, −2.452886547333875, −1.605237114684097, −1.046938035301256, 0,
1.046938035301256, 1.605237114684097, 2.452886547333875, 3.041936789854456, 3.563091082902004, 4.167699998049387, 4.722988536643146, 5.215779702216174, 5.699123116179744, 6.349073337121515, 6.939619144886551, 7.339663930455842, 7.934219308541970, 8.352918390722232, 8.930921114032481, 9.682320050318862, 9.915285047549310, 10.86972167807858, 11.01181703901586, 11.59299511860467, 11.88658741265851, 12.65695062139560, 12.99093937289402, 13.23980352982162, 14.08572393650125
