| L(s) = 1 | − 5-s − 3·9-s + 11-s − 7·17-s + 19-s − 2·23-s + 25-s − 8·29-s + 2·31-s − 9·37-s − 41-s + 9·43-s + 3·45-s − 7·47-s − 7·49-s − 12·53-s − 55-s − 10·59-s − 2·61-s + 13·67-s − 6·71-s − 4·73-s + 2·79-s + 9·81-s + 6·83-s + 7·85-s − 10·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 9-s + 0.301·11-s − 1.69·17-s + 0.229·19-s − 0.417·23-s + 1/5·25-s − 1.48·29-s + 0.359·31-s − 1.47·37-s − 0.156·41-s + 1.37·43-s + 0.447·45-s − 1.02·47-s − 49-s − 1.64·53-s − 0.134·55-s − 1.30·59-s − 0.256·61-s + 1.58·67-s − 0.712·71-s − 0.468·73-s + 0.225·79-s + 81-s + 0.658·83-s + 0.759·85-s − 1.05·89-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 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 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 9 T + p T^{2} \) | 1.37.j |
| 41 | \( 1 + T + p T^{2} \) | 1.41.b |
| 43 | \( 1 - 9 T + p T^{2} \) | 1.43.aj |
| 47 | \( 1 + 7 T + p T^{2} \) | 1.47.h |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 5 T + p T^{2} \) | 1.97.af |
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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
−13.73986969860516, −13.00517755428784, −12.66877342864618, −12.16519766419069, −11.49908341104463, −11.19546203621750, −11.01614835514328, −10.28714677309226, −9.627926137649797, −9.209942271635401, −8.688765656453644, −8.410991021141977, −7.679629611333311, −7.357055034422174, −6.601712335012140, −6.262162424801548, −5.735905629498103, −5.024207981014477, −4.604038685708915, −4.026835552762161, −3.286435796124033, −3.058491236004755, −1.972320876668814, −1.858396922007205, −0.5940652387542984, 0,
0.5940652387542984, 1.858396922007205, 1.972320876668814, 3.058491236004755, 3.286435796124033, 4.026835552762161, 4.604038685708915, 5.024207981014477, 5.735905629498103, 6.262162424801548, 6.601712335012140, 7.357055034422174, 7.679629611333311, 8.410991021141977, 8.688765656453644, 9.209942271635401, 9.627926137649797, 10.28714677309226, 11.01614835514328, 11.19546203621750, 11.49908341104463, 12.16519766419069, 12.66877342864618, 13.00517755428784, 13.73986969860516
