| L(s) = 1 | − 3-s − 5-s − 7-s + 9-s + 11-s + 13-s + 15-s + 5·19-s + 21-s − 2·23-s − 4·25-s − 27-s + 29-s − 8·31-s − 33-s + 35-s − 37-s − 39-s + 6·43-s − 45-s − 47-s + 49-s + 2·53-s − 55-s − 5·57-s + 9·59-s − 10·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.301·11-s + 0.277·13-s + 0.258·15-s + 1.14·19-s + 0.218·21-s − 0.417·23-s − 4/5·25-s − 0.192·27-s + 0.185·29-s − 1.43·31-s − 0.174·33-s + 0.169·35-s − 0.164·37-s − 0.160·39-s + 0.914·43-s − 0.149·45-s − 0.145·47-s + 1/7·49-s + 0.274·53-s − 0.134·55-s − 0.662·57-s + 1.17·59-s − 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 - T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + T + p T^{2} \) | 1.47.b |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 9 T + p T^{2} \) | 1.59.aj |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 9 T + p T^{2} \) | 1.73.aj |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−16.30078389865023, −15.88020958738295, −15.42542160577470, −14.71607159616520, −14.08329967195565, −13.56618738159777, −12.93703106444660, −12.26953777764382, −11.93252351866443, −11.25860340648381, −10.85142976841827, −10.08874380537432, −9.509153289505398, −9.048547662137284, −8.141084989003904, −7.622258928655864, −6.990750724258967, −6.400519701094185, −5.603188409921746, −5.263486193605581, −4.171322040950137, −3.798252371668573, −2.981724605302094, −1.945591165196182, −1.008502636699664, 0,
1.008502636699664, 1.945591165196182, 2.981724605302094, 3.798252371668573, 4.171322040950137, 5.263486193605581, 5.603188409921746, 6.400519701094185, 6.990750724258967, 7.622258928655864, 8.141084989003904, 9.048547662137284, 9.509153289505398, 10.08874380537432, 10.85142976841827, 11.25860340648381, 11.93252351866443, 12.26953777764382, 12.93703106444660, 13.56618738159777, 14.08329967195565, 14.71607159616520, 15.42542160577470, 15.88020958738295, 16.30078389865023
