| L(s) = 1 | + 2-s + 2·3-s + 4-s + 3·5-s + 2·6-s − 2·7-s + 8-s + 9-s + 3·10-s − 11-s + 2·12-s + 13-s − 2·14-s + 6·15-s + 16-s + 2·17-s + 18-s − 6·19-s + 3·20-s − 4·21-s − 22-s + 23-s + 2·24-s + 4·25-s + 26-s − 4·27-s − 2·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s + 1.34·5-s + 0.816·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.948·10-s − 0.301·11-s + 0.577·12-s + 0.277·13-s − 0.534·14-s + 1.54·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s − 1.37·19-s + 0.670·20-s − 0.872·21-s − 0.213·22-s + 0.208·23-s + 0.408·24-s + 4/5·25-s + 0.196·26-s − 0.769·27-s − 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| 239 | \( 1 + T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + T + p T^{2} \) | 1.47.b |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 15 T + p T^{2} \) | 1.59.p |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 - 9 T + p T^{2} \) | 1.73.aj |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 11 T + p T^{2} \) | 1.83.l |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−14.25637755261768, −13.88166029693654, −13.49630323647593, −13.09724138613840, −12.60079191327416, −12.30804668119546, −11.28399038385648, −10.86427762754233, −10.33232993674784, −9.699489180005081, −9.406276605613782, −8.914055536273930, −8.253598414930603, −7.793380544211093, −7.071919287603613, −6.502023088809425, −5.980657960634301, −5.629086189769030, −4.941845333339036, −4.120477046172428, −3.636557949684640, −2.962123394995501, −2.518889805335542, −1.983555652962009, −1.355425438890045, 0,
1.355425438890045, 1.983555652962009, 2.518889805335542, 2.962123394995501, 3.636557949684640, 4.120477046172428, 4.941845333339036, 5.629086189769030, 5.980657960634301, 6.502023088809425, 7.071919287603613, 7.793380544211093, 8.253598414930603, 8.914055536273930, 9.406276605613782, 9.699489180005081, 10.33232993674784, 10.86427762754233, 11.28399038385648, 12.30804668119546, 12.60079191327416, 13.09724138613840, 13.49630323647593, 13.88166029693654, 14.25637755261768
