| L(s) = 1 | + 2-s + 4-s + 8-s − 6·13-s + 16-s + 6·17-s − 6·23-s − 6·26-s + 4·31-s + 32-s + 6·34-s + 2·37-s − 12·41-s + 12·43-s − 6·46-s + 6·47-s − 7·49-s − 6·52-s + 6·53-s + 4·62-s + 64-s + 14·67-s + 6·68-s − 6·73-s + 2·74-s − 12·79-s − 12·82-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 1.66·13-s + 1/4·16-s + 1.45·17-s − 1.25·23-s − 1.17·26-s + 0.718·31-s + 0.176·32-s + 1.02·34-s + 0.328·37-s − 1.87·41-s + 1.82·43-s − 0.884·46-s + 0.875·47-s − 49-s − 0.832·52-s + 0.824·53-s + 0.508·62-s + 1/8·64-s + 1.71·67-s + 0.727·68-s − 0.702·73-s + 0.232·74-s − 1.35·79-s − 1.32·82-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−14.53110095641540, −14.10485287553271, −13.95591308877725, −12.99572895363944, −12.67597610397206, −12.10582359791386, −11.81710658748060, −11.34577418530119, −10.38232444673176, −10.12091841914436, −9.754649224937193, −9.021928138184290, −8.239303154927696, −7.752809567218756, −7.350213685393551, −6.722680933152030, −6.103892198055056, −5.402031725102612, −5.187530950491740, −4.314773974452775, −3.945634057516035, −3.069305252736910, −2.604602322227208, −1.919082902949382, −1.048213277270329, 0,
1.048213277270329, 1.919082902949382, 2.604602322227208, 3.069305252736910, 3.945634057516035, 4.314773974452775, 5.187530950491740, 5.402031725102612, 6.103892198055056, 6.722680933152030, 7.350213685393551, 7.752809567218756, 8.239303154927696, 9.021928138184290, 9.754649224937193, 10.12091841914436, 10.38232444673176, 11.34577418530119, 11.81710658748060, 12.10582359791386, 12.67597610397206, 12.99572895363944, 13.95591308877725, 14.10485287553271, 14.53110095641540
