| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 7-s + 8-s + 9-s − 10-s − 2·11-s + 12-s + 13-s + 14-s − 15-s + 16-s + 4·17-s + 18-s − 19-s − 20-s + 21-s − 2·22-s − 4·23-s + 24-s − 4·25-s + 26-s + 27-s + 28-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 + 1/3·9-s − 0.316·10-s − 0.603·11-s + 0.288·12-s + 0.277·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.970·17-s + 0.235·18-s − 0.229·19-s − 0.223·20-s + 0.218·21-s − 0.426·22-s − 0.834·23-s + 0.204·24-s − 4/5·25-s + 0.196·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 229242 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229242 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.965663373\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.965663373\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 3 | \( 1 - T \) | |
| 13 | \( 1 - T \) | |
| 2939 | \( 1 + T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 11 T + p T^{2} \) | 1.59.al |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 9 T + p T^{2} \) | 1.67.j |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−13.06332274747463, −12.58216756113784, −11.99269953607202, −11.61504324077389, −11.26242715663307, −10.57851948193236, −10.19140815825233, −9.797046973444926, −9.163343044742833, −8.480811792439937, −8.188299119037517, −7.730994998490830, −7.309616201036667, −6.794645642782514, −6.147740672664163, −5.570176844830925, −5.234641051612948, −4.610323027950740, −4.009165074071639, −3.627117537198603, −3.144306007879005, −2.555604209141758, −1.704999741544391, −1.593185388507559, −0.3719564527485419,
0.3719564527485419, 1.593185388507559, 1.704999741544391, 2.555604209141758, 3.144306007879005, 3.627117537198603, 4.009165074071639, 4.610323027950740, 5.234641051612948, 5.570176844830925, 6.147740672664163, 6.794645642782514, 7.309616201036667, 7.730994998490830, 8.188299119037517, 8.480811792439937, 9.163343044742833, 9.797046973444926, 10.19140815825233, 10.57851948193236, 11.26242715663307, 11.61504324077389, 11.99269953607202, 12.58216756113784, 13.06332274747463
