| L(s) = 1 | + 5-s + 5·7-s + 2·11-s − 13-s + 3·17-s − 2·19-s − 4·23-s − 4·25-s + 6·29-s − 4·31-s + 5·35-s + 11·37-s − 8·41-s − 43-s − 9·47-s + 18·49-s + 12·53-s + 2·55-s − 6·59-s − 65-s + 6·67-s − 7·71-s − 2·73-s + 10·77-s + 12·79-s + 16·83-s + 3·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.88·7-s + 0.603·11-s − 0.277·13-s + 0.727·17-s − 0.458·19-s − 0.834·23-s − 4/5·25-s + 1.11·29-s − 0.718·31-s + 0.845·35-s + 1.80·37-s − 1.24·41-s − 0.152·43-s − 1.31·47-s + 18/7·49-s + 1.64·53-s + 0.269·55-s − 0.781·59-s − 0.124·65-s + 0.733·67-s − 0.830·71-s − 0.234·73-s + 1.13·77-s + 1.35·79-s + 1.75·83-s + 0.325·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.141130990\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.141130990\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 - 5 T + p T^{2} \) | 1.7.af |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 11 T + p T^{2} \) | 1.37.al |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 6 T + p T^{2} \) | 1.67.ag |
| 71 | \( 1 + 7 T + p T^{2} \) | 1.71.h |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−10.10224008136720573625169806871, −9.225887962883235463810272968886, −8.190620854395079261439572770271, −7.81593597663044282712186867044, −6.60735802802977250427022700463, −5.61571239171975099148075759195, −4.80778500236897303143309002739, −3.91100411636941413859503964078, −2.29076143224509280614694719959, −1.35851693418797878688327656091,
1.35851693418797878688327656091, 2.29076143224509280614694719959, 3.91100411636941413859503964078, 4.80778500236897303143309002739, 5.61571239171975099148075759195, 6.60735802802977250427022700463, 7.81593597663044282712186867044, 8.190620854395079261439572770271, 9.225887962883235463810272968886, 10.10224008136720573625169806871
