| L(s) = 1 | + 3-s − 5·7-s + 9-s + 5·11-s + 19-s − 5·21-s − 8·23-s + 27-s + 3·29-s + 8·31-s + 5·33-s − 3·37-s − 5·41-s + 4·43-s − 47-s + 18·49-s + 53-s + 57-s + 6·59-s + 4·61-s − 5·63-s − 2·67-s − 8·69-s − 8·71-s − 17·73-s − 25·77-s − 10·79-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.88·7-s + 1/3·9-s + 1.50·11-s + 0.229·19-s − 1.09·21-s − 1.66·23-s + 0.192·27-s + 0.557·29-s + 1.43·31-s + 0.870·33-s − 0.493·37-s − 0.780·41-s + 0.609·43-s − 0.145·47-s + 18/7·49-s + 0.137·53-s + 0.132·57-s + 0.781·59-s + 0.512·61-s − 0.629·63-s − 0.244·67-s − 0.963·69-s − 0.949·71-s − 1.98·73-s − 2.84·77-s − 1.12·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 86700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 17 | \( 1 \) | |
| good | 7 | \( 1 + 5 T + p T^{2} \) | 1.7.f |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + T + p T^{2} \) | 1.47.b |
| 53 | \( 1 - T + p T^{2} \) | 1.53.ab |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 17 T + p T^{2} \) | 1.73.r |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−14.22541569070282, −13.66346432854369, −13.25991773507476, −12.64229469276340, −12.25963466534783, −11.76818273635517, −11.40481893572894, −10.28332406476128, −10.00030122394284, −9.868102502326359, −9.111975501893397, −8.684535435019409, −8.334733443650692, −7.370634671117471, −7.041175109469325, −6.472084087168339, −6.068531684602004, −5.637269433577287, −4.423225537479012, −4.209103886885274, −3.492634137949953, −3.109584054696884, −2.480199833628796, −1.669827999290742, −0.8943731222478847, 0,
0.8943731222478847, 1.669827999290742, 2.480199833628796, 3.109584054696884, 3.492634137949953, 4.209103886885274, 4.423225537479012, 5.637269433577287, 6.068531684602004, 6.472084087168339, 7.041175109469325, 7.370634671117471, 8.334733443650692, 8.684535435019409, 9.111975501893397, 9.868102502326359, 10.00030122394284, 10.28332406476128, 11.40481893572894, 11.76818273635517, 12.25963466534783, 12.64229469276340, 13.25991773507476, 13.66346432854369, 14.22541569070282
