| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s − 7-s + 8-s − 2·9-s − 10-s + 11-s − 12-s − 14-s + 15-s + 16-s − 17-s − 2·18-s − 3·19-s − 20-s + 21-s + 22-s + 6·23-s − 24-s + 25-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 + 0.301·11-s − 0.288·12-s − 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.471·18-s − 0.688·19-s − 0.223·20-s + 0.218·21-s + 0.213·22-s + 1.25·23-s − 0.204·24-s + 1/5·25-s + 0.962·27-s − 0.188·28-s − 0.185·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18590 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18590 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 17 | \( 1 + T + p T^{2} \) | 1.17.b |
| 19 | \( 1 + 3 T + p T^{2} \) | 1.19.d |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 5 T + p T^{2} \) | 1.53.f |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 - 13 T + p T^{2} \) | 1.61.an |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 5 T + p T^{2} \) | 1.71.af |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 + 17 T + p T^{2} \) | 1.89.r |
| 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
−15.93719778863779, −15.50246930041814, −14.89345945393793, −14.43388800838069, −13.88563158406613, −13.08523543992802, −12.86030343514522, −12.10234403333760, −11.60774155834866, −11.29319614518475, −10.60237035333413, −10.11669324521576, −9.274255303207095, −8.549445669395077, −8.201921332055129, −7.179128420303102, −6.685001064501739, −6.295402445798526, −5.469421474045243, −4.963333910151597, −4.346415732850317, −3.502030937519992, −3.001333225218146, −2.150735436435644, −1.028404909090640, 0,
1.028404909090640, 2.150735436435644, 3.001333225218146, 3.502030937519992, 4.346415732850317, 4.963333910151597, 5.469421474045243, 6.295402445798526, 6.685001064501739, 7.179128420303102, 8.201921332055129, 8.549445669395077, 9.274255303207095, 10.11669324521576, 10.60237035333413, 11.29319614518475, 11.60774155834866, 12.10234403333760, 12.86030343514522, 13.08523543992802, 13.88563158406613, 14.43388800838069, 14.89345945393793, 15.50246930041814, 15.93719778863779
