| L(s) = 1 | + 2·2-s + 2·4-s + 3·5-s − 3·9-s + 6·10-s + 3·11-s − 4·16-s + 3·17-s − 6·18-s + 19-s + 6·20-s + 6·22-s − 8·23-s + 4·25-s + 2·29-s + 2·31-s − 8·32-s + 6·34-s − 6·36-s − 4·37-s + 2·38-s − 6·41-s − 5·43-s + 6·44-s − 9·45-s − 16·46-s − 13·47-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 1.34·5-s − 9-s + 1.89·10-s + 0.904·11-s − 16-s + 0.727·17-s − 1.41·18-s + 0.229·19-s + 1.34·20-s + 1.27·22-s − 1.66·23-s + 4/5·25-s + 0.371·29-s + 0.359·31-s − 1.41·32-s + 1.02·34-s − 36-s − 0.657·37-s + 0.324·38-s − 0.937·41-s − 0.762·43-s + 0.904·44-s − 1.34·45-s − 2.35·46-s − 1.89·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49343 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49343 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) | |
| 19 | \( 1 - T \) | |
| 53 | \( 1 + T \) | |
| good | 2 | \( 1 - p T + p T^{2} \) | 1.2.ac |
| 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 + 13 T + p T^{2} \) | 1.47.n |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 16 T + p T^{2} \) | 1.89.q |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−14.45125126522968, −14.29124128789160, −13.80687370802233, −13.44308818457656, −12.90855313437283, −12.24974880876372, −11.74059489525096, −11.62437201412642, −10.76511074855956, −10.01802263137227, −9.732743233778806, −9.178582638859424, −8.353277606297797, −8.158891202590528, −6.901972425110018, −6.592141696322275, −6.061691051793395, −5.567124065151933, −5.222201192134486, −4.568133253540339, −3.667675640341651, −3.412691789360099, −2.555924097808153, −2.039628968360577, −1.306234493152241, 0,
1.306234493152241, 2.039628968360577, 2.555924097808153, 3.412691789360099, 3.667675640341651, 4.568133253540339, 5.222201192134486, 5.567124065151933, 6.061691051793395, 6.592141696322275, 6.901972425110018, 8.158891202590528, 8.353277606297797, 9.178582638859424, 9.732743233778806, 10.01802263137227, 10.76511074855956, 11.62437201412642, 11.74059489525096, 12.24974880876372, 12.90855313437283, 13.44308818457656, 13.80687370802233, 14.29124128789160, 14.45125126522968
