| L(s) = 1 | + 3·5-s + 7-s − 4·11-s − 13-s + 7·17-s + 6·19-s − 6·23-s + 4·25-s + 8·31-s + 3·35-s + 9·37-s + 6·41-s + 5·43-s − 5·47-s − 6·49-s + 6·53-s − 12·55-s + 2·61-s − 3·65-s − 2·67-s + 9·71-s + 8·73-s − 4·77-s − 8·79-s − 4·83-s + 21·85-s − 14·89-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 0.377·7-s − 1.20·11-s − 0.277·13-s + 1.69·17-s + 1.37·19-s − 1.25·23-s + 4/5·25-s + 1.43·31-s + 0.507·35-s + 1.47·37-s + 0.937·41-s + 0.762·43-s − 0.729·47-s − 6/7·49-s + 0.824·53-s − 1.61·55-s + 0.256·61-s − 0.372·65-s − 0.244·67-s + 1.06·71-s + 0.936·73-s − 0.455·77-s − 0.900·79-s − 0.439·83-s + 2.27·85-s − 1.48·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14976 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14976 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.268063519\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.268063519\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 9 T + p T^{2} \) | 1.37.aj |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 + 5 T + p T^{2} \) | 1.47.f |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 16 T + p T^{2} \) | 1.97.q |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.21079786536661, −15.56760553029915, −14.77075335135320, −14.32076304902533, −13.71771800661237, −13.55963693071292, −12.59883435998315, −12.32791580718009, −11.51095259733770, −10.94494559991593, −10.08315792897085, −9.767200669723551, −9.629781779034453, −8.434327340232022, −7.883938916352921, −7.548818237791853, −6.572931502991851, −5.736379801905322, −5.585850511182600, −4.907347360884272, −4.040334757449199, −2.914598466332713, −2.584315979260942, −1.605065120568284, −0.8259695144944730,
0.8259695144944730, 1.605065120568284, 2.584315979260942, 2.914598466332713, 4.040334757449199, 4.907347360884272, 5.585850511182600, 5.736379801905322, 6.572931502991851, 7.548818237791853, 7.883938916352921, 8.434327340232022, 9.629781779034453, 9.767200669723551, 10.08315792897085, 10.94494559991593, 11.51095259733770, 12.32791580718009, 12.59883435998315, 13.55963693071292, 13.71771800661237, 14.32076304902533, 14.77075335135320, 15.56760553029915, 16.21079786536661
