| L(s) = 1 | + 3-s + 5·7-s − 2·9-s − 4·11-s + 3·13-s − 17-s + 2·19-s + 5·21-s + 8·23-s − 5·27-s − 5·31-s − 4·33-s − 12·37-s + 3·39-s − 10·41-s − 4·43-s + 2·47-s + 18·49-s − 51-s − 53-s + 2·57-s − 2·61-s − 10·63-s + 8·67-s + 8·69-s + 5·71-s − 4·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.88·7-s − 2/3·9-s − 1.20·11-s + 0.832·13-s − 0.242·17-s + 0.458·19-s + 1.09·21-s + 1.66·23-s − 0.962·27-s − 0.898·31-s − 0.696·33-s − 1.97·37-s + 0.480·39-s − 1.56·41-s − 0.609·43-s + 0.291·47-s + 18/7·49-s − 0.140·51-s − 0.137·53-s + 0.264·57-s − 0.256·61-s − 1.25·63-s + 0.977·67-s + 0.963·69-s + 0.593·71-s − 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 7 | \( 1 - 5 T + p T^{2} \) | 1.7.af |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 + 12 T + p T^{2} \) | 1.37.m |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 5 T + p T^{2} \) | 1.71.af |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 17 T + p T^{2} \) | 1.79.r |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 16 T + p T^{2} \) | 1.97.q |
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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
−15.32238095167044, −15.11275366508723, −14.27072676432229, −14.13334586167285, −13.44529257835946, −13.08702893679859, −12.28196995983180, −11.54350335029729, −11.27263878693905, −10.72451372611188, −10.32654545411398, −9.340896517097435, −8.733860036548224, −8.358533467351498, −8.084586107382808, −7.231845152003068, −6.908316135773178, −5.622477793993295, −5.348677131240588, −4.913193485846138, −4.041529536772330, −3.257604215963050, −2.698900623277370, −1.826203940179685, −1.325042110166912, 0,
1.325042110166912, 1.826203940179685, 2.698900623277370, 3.257604215963050, 4.041529536772330, 4.913193485846138, 5.348677131240588, 5.622477793993295, 6.908316135773178, 7.231845152003068, 8.084586107382808, 8.358533467351498, 8.733860036548224, 9.340896517097435, 10.32654545411398, 10.72451372611188, 11.27263878693905, 11.54350335029729, 12.28196995983180, 13.08702893679859, 13.44529257835946, 14.13334586167285, 14.27072676432229, 15.11275366508723, 15.32238095167044
