| L(s) = 1 | − 2-s + 4-s − 7-s − 8-s + 5·13-s + 14-s + 16-s + 3·17-s − 5·19-s − 5·26-s − 28-s − 3·29-s + 2·31-s − 32-s − 3·34-s − 5·37-s + 5·38-s + 6·41-s + 2·43-s − 6·47-s − 6·49-s + 5·52-s + 56-s + 3·58-s − 8·61-s − 2·62-s + 64-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s + 1.38·13-s + 0.267·14-s + 1/4·16-s + 0.727·17-s − 1.14·19-s − 0.980·26-s − 0.188·28-s − 0.557·29-s + 0.359·31-s − 0.176·32-s − 0.514·34-s − 0.821·37-s + 0.811·38-s + 0.937·41-s + 0.304·43-s − 0.875·47-s − 6/7·49-s + 0.693·52-s + 0.133·56-s + 0.393·58-s − 1.02·61-s − 0.254·62-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 - 3 T + p T^{2} \) | 1.83.ad |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−14.69089978329004, −14.25216395903991, −13.63802430065681, −13.02726744788964, −12.69706363643322, −12.11699286857389, −11.35406950943299, −11.16566869287047, −10.40357597651469, −10.17216192604852, −9.465774971793058, −8.835632261264738, −8.634149688481596, −7.871763475208989, −7.502040823124428, −6.731694866518283, −6.098118199857170, −6.014256875176563, −5.045151686324946, −4.357768439722537, −3.548699833284772, −3.247158497998582, −2.297223198626108, −1.637655954788831, −0.9195899165097771, 0,
0.9195899165097771, 1.637655954788831, 2.297223198626108, 3.247158497998582, 3.548699833284772, 4.357768439722537, 5.045151686324946, 6.014256875176563, 6.098118199857170, 6.731694866518283, 7.502040823124428, 7.871763475208989, 8.634149688481596, 8.835632261264738, 9.465774971793058, 10.17216192604852, 10.40357597651469, 11.16566869287047, 11.35406950943299, 12.11699286857389, 12.69706363643322, 13.02726744788964, 13.63802430065681, 14.25216395903991, 14.69089978329004
