| L(s) = 1 | − 5-s − 3·9-s − 11-s + 2·17-s + 4·19-s − 8·23-s + 25-s + 10·29-s + 8·31-s − 6·37-s + 2·41-s + 12·43-s + 3·45-s − 4·47-s − 7·49-s − 6·53-s + 55-s − 4·59-s + 10·61-s + 4·67-s + 14·73-s + 8·79-s + 9·81-s + 12·83-s − 2·85-s + 14·89-s − 4·95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 9-s − 0.301·11-s + 0.485·17-s + 0.917·19-s − 1.66·23-s + 1/5·25-s + 1.85·29-s + 1.43·31-s − 0.986·37-s + 0.312·41-s + 1.82·43-s + 0.447·45-s − 0.583·47-s − 49-s − 0.824·53-s + 0.134·55-s − 0.520·59-s + 1.28·61-s + 0.488·67-s + 1.63·73-s + 0.900·79-s + 81-s + 1.31·83-s − 0.216·85-s + 1.48·89-s − 0.410·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 148720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 148720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.136081647\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.136081647\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−13.51982256688355, −12.73769113061811, −12.22863531865227, −11.93968788907684, −11.63300357186968, −10.88417530285558, −10.59558771008572, −9.932063209003698, −9.588249523708655, −8.991031942451566, −8.276673281145489, −8.013692764201187, −7.802505252629315, −6.883156301266847, −6.443347513833575, −5.947260651676360, −5.401807910875067, −4.816295128243780, −4.401174020034176, −3.500860464782574, −3.271074585490467, −2.537628705308352, −2.041113806704021, −0.9734268219110378, −0.5195003724517325,
0.5195003724517325, 0.9734268219110378, 2.041113806704021, 2.537628705308352, 3.271074585490467, 3.500860464782574, 4.401174020034176, 4.816295128243780, 5.401807910875067, 5.947260651676360, 6.443347513833575, 6.883156301266847, 7.802505252629315, 8.013692764201187, 8.276673281145489, 8.991031942451566, 9.588249523708655, 9.932063209003698, 10.59558771008572, 10.88417530285558, 11.63300357186968, 11.93968788907684, 12.22863531865227, 12.73769113061811, 13.51982256688355
