| L(s) = 1 | + 2-s + 4-s + 2·5-s − 2·7-s + 8-s + 2·10-s − 2·11-s + 4·13-s − 2·14-s + 16-s + 4·17-s + 4·19-s + 2·20-s − 2·22-s − 6·23-s − 25-s + 4·26-s − 2·28-s − 4·31-s + 32-s + 4·34-s − 4·35-s + 37-s + 4·38-s + 2·40-s + 10·41-s − 2·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.755·7-s + 0.353·8-s + 0.632·10-s − 0.603·11-s + 1.10·13-s − 0.534·14-s + 1/4·16-s + 0.970·17-s + 0.917·19-s + 0.447·20-s − 0.426·22-s − 1.25·23-s − 1/5·25-s + 0.784·26-s − 0.377·28-s − 0.718·31-s + 0.176·32-s + 0.685·34-s − 0.676·35-s + 0.164·37-s + 0.648·38-s + 0.316·40-s + 1.56·41-s − 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136242 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136242 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 29 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 11 T + p T^{2} \) | 1.59.l |
| 61 | \( 1 - 11 T + p T^{2} \) | 1.61.al |
| 67 | \( 1 + 11 T + p T^{2} \) | 1.67.l |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 11 T + p T^{2} \) | 1.79.al |
| 83 | \( 1 - 11 T + p T^{2} \) | 1.83.al |
| 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
−13.56329892729915, −13.41682377236257, −12.70952161474832, −12.40107706699359, −11.94648519743386, −11.20111894626173, −10.87641917715838, −10.31051025443932, −9.776499984580390, −9.502105806904833, −8.982977467277813, −8.061964857471973, −7.845303168195763, −7.250890155359011, −6.490091413796746, −6.119777737651945, −5.678050689104468, −5.421096310044640, −4.618578992563491, −3.939523655837021, −3.460860194933480, −2.952879804017653, −2.346798185235930, −1.632692915754031, −1.071490788083813, 0,
1.071490788083813, 1.632692915754031, 2.346798185235930, 2.952879804017653, 3.460860194933480, 3.939523655837021, 4.618578992563491, 5.421096310044640, 5.678050689104468, 6.119777737651945, 6.490091413796746, 7.250890155359011, 7.845303168195763, 8.061964857471973, 8.982977467277813, 9.502105806904833, 9.776499984580390, 10.31051025443932, 10.87641917715838, 11.20111894626173, 11.94648519743386, 12.40107706699359, 12.70952161474832, 13.41682377236257, 13.56329892729915
