| L(s) = 1 | − 2-s − 3-s + 4-s − 4·5-s + 6-s − 8-s + 9-s + 4·10-s − 2·11-s − 12-s + 4·15-s + 16-s − 18-s + 2·19-s − 4·20-s + 2·22-s + 24-s + 11·25-s − 27-s + 2·29-s − 4·30-s − 8·31-s − 32-s + 2·33-s + 36-s + 10·37-s − 2·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.78·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 1.26·10-s − 0.603·11-s − 0.288·12-s + 1.03·15-s + 1/4·16-s − 0.235·18-s + 0.458·19-s − 0.894·20-s + 0.426·22-s + 0.204·24-s + 11/5·25-s − 0.192·27-s + 0.371·29-s − 0.730·30-s − 1.43·31-s − 0.176·32-s + 0.348·33-s + 1/6·36-s + 1.64·37-s − 0.324·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 293046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 293046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5481733113\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5481733113\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 12 T + p T^{2} \) | 1.41.am |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−12.63551030739459, −12.09746707715382, −11.61196641689941, −11.36223815769431, −10.88508902339282, −10.65910906501495, −9.986002213390019, −9.383114252680359, −9.114546964979719, −8.390910885672851, −7.926525407253150, −7.626935034740154, −7.387379675938688, −6.700386891628634, −6.225403965053246, −5.686784127467471, −4.961683647130165, −4.638608366751518, −4.015344208760030, −3.511250255958828, −2.969372159512404, −2.395995707742725, −1.511724513133793, −0.8269238351958147, −0.3174130859052060,
0.3174130859052060, 0.8269238351958147, 1.511724513133793, 2.395995707742725, 2.969372159512404, 3.511250255958828, 4.015344208760030, 4.638608366751518, 4.961683647130165, 5.686784127467471, 6.225403965053246, 6.700386891628634, 7.387379675938688, 7.626935034740154, 7.926525407253150, 8.390910885672851, 9.114546964979719, 9.383114252680359, 9.986002213390019, 10.65910906501495, 10.88508902339282, 11.36223815769431, 11.61196641689941, 12.09746707715382, 12.63551030739459
