| L(s) = 1 | − 2·3-s + 5-s + 7-s + 9-s + 2·13-s − 2·15-s + 6·17-s − 2·19-s − 2·21-s + 6·23-s + 25-s + 4·27-s − 8·31-s + 35-s + 4·37-s − 4·39-s − 12·41-s + 4·43-s + 45-s − 12·47-s + 49-s − 12·51-s + 4·57-s + 2·61-s + 63-s + 2·65-s + 8·67-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.554·13-s − 0.516·15-s + 1.45·17-s − 0.458·19-s − 0.436·21-s + 1.25·23-s + 1/5·25-s + 0.769·27-s − 1.43·31-s + 0.169·35-s + 0.657·37-s − 0.640·39-s − 1.87·41-s + 0.609·43-s + 0.149·45-s − 1.75·47-s + 1/7·49-s − 1.68·51-s + 0.529·57-s + 0.256·61-s + 0.125·63-s + 0.248·65-s + 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 271040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 271040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−12.95260946851092, −12.54061770699606, −11.96434748057232, −11.64554610912362, −11.10050296722067, −10.84307592122228, −10.37005329225306, −9.869827099023764, −9.421104523625567, −8.780447589958400, −8.426430254460365, −7.836498443427459, −7.289940644050378, −6.760222880031073, −6.337269816705729, −5.838469656214438, −5.325836172191551, −5.114536782981972, −4.560962440707362, −3.772951414094325, −3.293943420143253, −2.733475296139695, −1.842090666933771, −1.369263089896060, −0.8006823502467607, 0,
0.8006823502467607, 1.369263089896060, 1.842090666933771, 2.733475296139695, 3.293943420143253, 3.772951414094325, 4.560962440707362, 5.114536782981972, 5.325836172191551, 5.838469656214438, 6.337269816705729, 6.760222880031073, 7.289940644050378, 7.836498443427459, 8.426430254460365, 8.780447589958400, 9.421104523625567, 9.869827099023764, 10.37005329225306, 10.84307592122228, 11.10050296722067, 11.64554610912362, 11.96434748057232, 12.54061770699606, 12.95260946851092
