| L(s) = 1 | − 3-s − 7-s − 2·9-s − 4·11-s + 7·13-s − 17-s − 6·19-s + 21-s + 8·23-s + 5·27-s − 7·31-s + 4·33-s − 8·37-s − 7·39-s − 6·41-s − 4·43-s + 6·47-s − 6·49-s + 51-s + 3·53-s + 6·57-s − 12·59-s + 14·61-s + 2·63-s − 8·67-s − 8·69-s − 9·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s − 2/3·9-s − 1.20·11-s + 1.94·13-s − 0.242·17-s − 1.37·19-s + 0.218·21-s + 1.66·23-s + 0.962·27-s − 1.25·31-s + 0.696·33-s − 1.31·37-s − 1.12·39-s − 0.937·41-s − 0.609·43-s + 0.875·47-s − 6/7·49-s + 0.140·51-s + 0.412·53-s + 0.794·57-s − 1.56·59-s + 1.79·61-s + 0.251·63-s − 0.977·67-s − 0.963·69-s − 1.06·71-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
| \(L(1)\) |
\(\approx\) |
\(0.7399016440\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7399016440\) |
| \(L(\frac{3}{2})\) |
|
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.b |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 7 T + p T^{2} \) | 1.13.ah |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 5 T + p T^{2} \) | 1.79.af |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−15.45159000413068, −14.81981135175970, −14.16831929090625, −13.42103728092114, −13.15531352745540, −12.73899799081231, −12.03034750038923, −11.27570669407989, −10.93740296413240, −10.61380438044054, −10.05349895157459, −8.963017073087170, −8.707954157822306, −8.350308508411971, −7.384027935652538, −6.823708724566908, −6.185519142610234, −5.766037932882156, −5.133171397479975, −4.554631499831758, −3.516429243062097, −3.216086709491080, −2.279824469446508, −1.415700907544100, −0.3513578601424908,
0.3513578601424908, 1.415700907544100, 2.279824469446508, 3.216086709491080, 3.516429243062097, 4.554631499831758, 5.133171397479975, 5.766037932882156, 6.185519142610234, 6.823708724566908, 7.384027935652538, 8.350308508411971, 8.707954157822306, 8.963017073087170, 10.05349895157459, 10.61380438044054, 10.93740296413240, 11.27570669407989, 12.03034750038923, 12.73899799081231, 13.15531352745540, 13.42103728092114, 14.16831929090625, 14.81981135175970, 15.45159000413068
