| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 3·7-s + 8-s − 2·9-s + 10-s − 11-s − 12-s + 13-s − 3·14-s − 15-s + 16-s − 5·17-s − 2·18-s + 19-s + 20-s + 3·21-s − 22-s − 3·23-s − 24-s + 25-s + 26-s + 5·27-s − 3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 1.13·7-s + 0.353·8-s − 2/3·9-s + 0.316·10-s − 0.301·11-s − 0.288·12-s + 0.277·13-s − 0.801·14-s − 0.258·15-s + 1/4·16-s − 1.21·17-s − 0.471·18-s + 0.229·19-s + 0.223·20-s + 0.654·21-s − 0.213·22-s − 0.625·23-s − 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.962·27-s − 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 - T \) | |
| 13 | \( 1 - T \) | |
| 19 | \( 1 - T \) | |
| 29 | \( 1 - T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 11 T + p T^{2} \) | 1.53.l |
| 59 | \( 1 + 7 T + p T^{2} \) | 1.59.h |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−14.39725957092466, −14.08605312737244, −13.57677653319958, −13.05193527313820, −12.70443232035907, −12.25442092828750, −11.59454266849098, −11.19429337934907, −10.64893669969641, −10.28651163345582, −9.544071671544841, −9.158997150703372, −8.507124841437163, −7.969075071643077, −7.028984545966022, −6.750053780903296, −6.219773720858587, −5.792474700091242, −5.216069080791215, −4.776219488823027, −3.958349521527651, −3.259434152370544, −2.988484388394533, −2.056608878539332, −1.529112908477719, 0, 0,
1.529112908477719, 2.056608878539332, 2.988484388394533, 3.259434152370544, 3.958349521527651, 4.776219488823027, 5.216069080791215, 5.792474700091242, 6.219773720858587, 6.750053780903296, 7.028984545966022, 7.969075071643077, 8.507124841437163, 9.158997150703372, 9.544071671544841, 10.28651163345582, 10.64893669969641, 11.19429337934907, 11.59454266849098, 12.25442092828750, 12.70443232035907, 13.05193527313820, 13.57677653319958, 14.08605312737244, 14.39725957092466
