| L(s) = 1 | − 2·5-s − 2·7-s + 13-s − 2·17-s + 4·23-s − 25-s + 2·29-s − 2·31-s + 4·35-s + 6·37-s + 6·41-s − 2·43-s + 6·47-s − 3·49-s − 6·53-s − 8·59-s − 2·61-s − 2·65-s − 8·67-s − 2·71-s − 14·73-s − 4·79-s + 8·83-s + 4·85-s + 14·89-s − 2·91-s + 2·97-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.755·7-s + 0.277·13-s − 0.485·17-s + 0.834·23-s − 1/5·25-s + 0.371·29-s − 0.359·31-s + 0.676·35-s + 0.986·37-s + 0.937·41-s − 0.304·43-s + 0.875·47-s − 3/7·49-s − 0.824·53-s − 1.04·59-s − 0.256·61-s − 0.248·65-s − 0.977·67-s − 0.237·71-s − 1.63·73-s − 0.450·79-s + 0.878·83-s + 0.433·85-s + 1.48·89-s − 0.209·91-s + 0.203·97-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−16.11951978821979, −15.88360462340310, −15.30257868632695, −14.78621146290608, −14.13493529687967, −13.45981335121479, −12.87406697487388, −12.57627390958203, −11.63500031845346, −11.47377694653702, −10.65321086718577, −10.20614646114651, −9.272337755834083, −9.032938280047649, −8.236945547281039, −7.521492564671700, −7.205597636053604, −6.186620230023898, −6.014980129802505, −4.788613792331212, −4.409003557902065, −3.500358767581996, −3.079022974555756, −2.117096587545115, −0.9553458056022991, 0,
0.9553458056022991, 2.117096587545115, 3.079022974555756, 3.500358767581996, 4.409003557902065, 4.788613792331212, 6.014980129802505, 6.186620230023898, 7.205597636053604, 7.521492564671700, 8.236945547281039, 9.032938280047649, 9.272337755834083, 10.20614646114651, 10.65321086718577, 11.47377694653702, 11.63500031845346, 12.57627390958203, 12.87406697487388, 13.45981335121479, 14.13493529687967, 14.78621146290608, 15.30257868632695, 15.88360462340310, 16.11951978821979
