| L(s) = 1 | − 7-s − 4·11-s + 6·13-s − 6·19-s + 23-s − 5·25-s + 3·29-s + 3·37-s + 6·41-s − 7·43-s − 12·47-s + 49-s + 3·53-s + 10·59-s + 2·61-s − 4·67-s + 3·71-s − 8·73-s + 4·77-s − 7·79-s + 6·83-s + 10·89-s − 6·91-s − 8·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 1.20·11-s + 1.66·13-s − 1.37·19-s + 0.208·23-s − 25-s + 0.557·29-s + 0.493·37-s + 0.937·41-s − 1.06·43-s − 1.75·47-s + 1/7·49-s + 0.412·53-s + 1.30·59-s + 0.256·61-s − 0.488·67-s + 0.356·71-s − 0.936·73-s + 0.455·77-s − 0.787·79-s + 0.658·83-s + 1.05·89-s − 0.628·91-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72828 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72828 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 + 7 T + p T^{2} \) | 1.79.h |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−14.36804081552139, −13.63661414795477, −13.27931689315272, −12.98805329083979, −12.55789195952370, −11.66239380697106, −11.40404365240738, −10.75522082948220, −10.36858693218368, −9.931831518457149, −9.260476340901755, −8.610808276932243, −8.261655169890581, −7.868011752513095, −7.058261052805514, −6.528004444561880, −5.992194460284102, −5.639296049447587, −4.811084147611218, −4.292854199052400, −3.614274702407847, −3.119505494185419, −2.347006050564409, −1.768645167511064, −0.8339412781908774, 0,
0.8339412781908774, 1.768645167511064, 2.347006050564409, 3.119505494185419, 3.614274702407847, 4.292854199052400, 4.811084147611218, 5.639296049447587, 5.992194460284102, 6.528004444561880, 7.058261052805514, 7.868011752513095, 8.261655169890581, 8.610808276932243, 9.260476340901755, 9.931831518457149, 10.36858693218368, 10.75522082948220, 11.40404365240738, 11.66239380697106, 12.55789195952370, 12.98805329083979, 13.27931689315272, 13.63661414795477, 14.36804081552139
