| L(s) = 1 | − 5-s − 7-s − 3·9-s + 2·11-s + 13-s − 7·19-s − 3·23-s − 4·25-s − 9·29-s + 5·31-s + 35-s − 8·37-s − 10·41-s + 5·43-s + 3·45-s + 7·47-s + 49-s + 3·53-s − 2·55-s + 6·61-s + 3·63-s − 65-s − 10·67-s + 4·71-s − 11·73-s − 2·77-s − 11·79-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s − 9-s + 0.603·11-s + 0.277·13-s − 1.60·19-s − 0.625·23-s − 4/5·25-s − 1.67·29-s + 0.898·31-s + 0.169·35-s − 1.31·37-s − 1.56·41-s + 0.762·43-s + 0.447·45-s + 1.02·47-s + 1/7·49-s + 0.412·53-s − 0.269·55-s + 0.768·61-s + 0.377·63-s − 0.124·65-s − 1.22·67-s + 0.474·71-s − 1.28·73-s − 0.227·77-s − 1.23·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 + 11 T + p T^{2} \) | 1.79.l |
| 83 | \( 1 - 11 T + p T^{2} \) | 1.83.al |
| 89 | \( 1 + 3 T + p T^{2} \) | 1.89.d |
| 97 | \( 1 + 15 T + p T^{2} \) | 1.97.p |
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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
−10.01513649753358836657539015140, −8.904130419685472930571593520552, −8.436689878182382611349679160524, −7.35663560732594657846546069334, −6.34340387335422307141360060611, −5.63592882191656149796503033584, −4.24611636549978990034128034887, −3.44651529114702159448438484724, −2.06843208978381721287541494629, 0,
2.06843208978381721287541494629, 3.44651529114702159448438484724, 4.24611636549978990034128034887, 5.63592882191656149796503033584, 6.34340387335422307141360060611, 7.35663560732594657846546069334, 8.436689878182382611349679160524, 8.904130419685472930571593520552, 10.01513649753358836657539015140
