| L(s) = 1 | + 5-s − 11-s − 4·13-s − 7·17-s + 19-s − 3·23-s + 25-s + 5·29-s + 2·31-s − 8·37-s − 6·41-s − 11·43-s − 10·47-s + 53-s − 55-s − 59-s − 61-s − 4·65-s − 6·67-s + 10·71-s − 16·73-s − 4·79-s + 9·83-s − 7·85-s − 15·89-s + 95-s + 5·97-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.301·11-s − 1.10·13-s − 1.69·17-s + 0.229·19-s − 0.625·23-s + 1/5·25-s + 0.928·29-s + 0.359·31-s − 1.31·37-s − 0.937·41-s − 1.67·43-s − 1.45·47-s + 0.137·53-s − 0.134·55-s − 0.130·59-s − 0.128·61-s − 0.496·65-s − 0.733·67-s + 1.18·71-s − 1.87·73-s − 0.450·79-s + 0.987·83-s − 0.759·85-s − 1.58·89-s + 0.102·95-s + 0.507·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 194040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 11 T + p T^{2} \) | 1.43.l |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 - T + p T^{2} \) | 1.53.ab |
| 59 | \( 1 + T + p T^{2} \) | 1.59.b |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 + 6 T + p T^{2} \) | 1.67.g |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 + 16 T + p T^{2} \) | 1.73.q |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 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
−13.46949625955135, −13.29411123103474, −12.58608029221581, −12.11400024324347, −11.77810563147012, −11.24268896309593, −10.64268857294224, −10.22536987858761, −9.842146816998907, −9.409222293636725, −8.732882784957167, −8.387915684890364, −7.954614282198630, −7.158070465865249, −6.774234765085983, −6.510438700351635, −5.747208700393573, −5.210808072067787, −4.695659517333464, −4.441617445395214, −3.532966905515418, −3.003141877640245, −2.401092122853985, −1.899838955705042, −1.323196145863170, 0, 0,
1.323196145863170, 1.899838955705042, 2.401092122853985, 3.003141877640245, 3.532966905515418, 4.441617445395214, 4.695659517333464, 5.210808072067787, 5.747208700393573, 6.510438700351635, 6.774234765085983, 7.158070465865249, 7.954614282198630, 8.387915684890364, 8.732882784957167, 9.409222293636725, 9.842146816998907, 10.22536987858761, 10.64268857294224, 11.24268896309593, 11.77810563147012, 12.11400024324347, 12.58608029221581, 13.29411123103474, 13.46949625955135
