| L(s) = 1 | − 3·5-s − 5·11-s + 4·17-s + 23-s + 4·25-s + 3·29-s + 9·31-s − 12·37-s − 12·41-s − 6·43-s + 2·47-s − 5·53-s + 15·55-s − 9·59-s − 2·61-s + 2·67-s + 6·71-s − 14·73-s − 11·79-s − 17·83-s − 12·85-s − 10·89-s − 13·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 1.50·11-s + 0.970·17-s + 0.208·23-s + 4/5·25-s + 0.557·29-s + 1.61·31-s − 1.97·37-s − 1.87·41-s − 0.914·43-s + 0.291·47-s − 0.686·53-s + 2.02·55-s − 1.17·59-s − 0.256·61-s + 0.244·67-s + 0.712·71-s − 1.63·73-s − 1.23·79-s − 1.86·83-s − 1.30·85-s − 1.05·89-s − 1.31·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 9 T + p T^{2} \) | 1.31.aj |
| 37 | \( 1 + 12 T + p T^{2} \) | 1.37.m |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 + 5 T + p T^{2} \) | 1.53.f |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 11 T + p T^{2} \) | 1.79.l |
| 83 | \( 1 + 17 T + p T^{2} \) | 1.83.r |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 13 T + p T^{2} \) | 1.97.n |
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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
−13.64629922765229, −13.37535102957205, −12.54117023986371, −12.35754801729050, −11.83359088982506, −11.52768656962550, −10.82013964513224, −10.43056906329424, −10.06252077352836, −9.570289792071275, −8.626250073542864, −8.299849318530916, −8.114301350734684, −7.490046061293148, −6.981501846225483, −6.637345976386022, −5.672392577191721, −5.372118679337146, −4.710312695510560, −4.379320854421261, −3.591897081703813, −2.963660447777502, −2.923360479262027, −1.776076768561802, −1.152744491264893, 0, 0,
1.152744491264893, 1.776076768561802, 2.923360479262027, 2.963660447777502, 3.591897081703813, 4.379320854421261, 4.710312695510560, 5.372118679337146, 5.672392577191721, 6.637345976386022, 6.981501846225483, 7.490046061293148, 8.114301350734684, 8.299849318530916, 8.626250073542864, 9.570289792071275, 10.06252077352836, 10.43056906329424, 10.82013964513224, 11.52768656962550, 11.83359088982506, 12.35754801729050, 12.54117023986371, 13.37535102957205, 13.64629922765229
