| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 7-s − 8-s − 2·9-s + 10-s − 12-s + 4·13-s + 14-s + 15-s + 16-s + 3·17-s + 2·18-s + 4·19-s − 20-s + 21-s + 4·23-s + 24-s − 4·25-s − 4·26-s + 5·27-s − 28-s + 29-s − 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s − 2/3·9-s + 0.316·10-s − 0.288·12-s + 1.10·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.727·17-s + 0.471·18-s + 0.917·19-s − 0.223·20-s + 0.218·21-s + 0.834·23-s + 0.204·24-s − 4/5·25-s − 0.784·26-s + 0.962·27-s − 0.188·28-s + 0.185·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23534 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23534 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 41 | \( 1 \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 43 | \( 1 + 9 T + p T^{2} \) | 1.43.j |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 - 16 T + p T^{2} \) | 1.73.aq |
| 79 | \( 1 - 11 T + p T^{2} \) | 1.79.al |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - T + p T^{2} \) | 1.89.ab |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
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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
−15.93688229022877, −15.24782491329702, −14.90618761036361, −13.96127480632336, −13.68146911756575, −12.92619401052608, −12.17276375745301, −11.87373759862019, −11.30161654000629, −10.85617950142725, −10.31990830011753, −9.639275709675927, −9.072745687289067, −8.543096408529161, −7.920649304351551, −7.437985194136184, −6.624863640068222, −6.229178438739196, −5.469516177147266, −5.092553769322587, −3.937511290164340, −3.351695451682221, −2.813602869021348, −1.634150554170338, −0.8892132547199924, 0,
0.8892132547199924, 1.634150554170338, 2.813602869021348, 3.351695451682221, 3.937511290164340, 5.092553769322587, 5.469516177147266, 6.229178438739196, 6.624863640068222, 7.437985194136184, 7.920649304351551, 8.543096408529161, 9.072745687289067, 9.639275709675927, 10.31990830011753, 10.85617950142725, 11.30161654000629, 11.87373759862019, 12.17276375745301, 12.92619401052608, 13.68146911756575, 13.96127480632336, 14.90618761036361, 15.24782491329702, 15.93688229022877
