| L(s) = 1 | − 2-s + 4-s + 5-s − 7-s − 8-s − 10-s + 5·11-s + 14-s + 16-s − 3·17-s + 7·19-s + 20-s − 5·22-s − 4·23-s + 25-s − 28-s − 5·29-s + 6·31-s − 32-s + 3·34-s − 35-s + 2·37-s − 7·38-s − 40-s + 3·41-s − 6·43-s + 5·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s − 0.353·8-s − 0.316·10-s + 1.50·11-s + 0.267·14-s + 1/4·16-s − 0.727·17-s + 1.60·19-s + 0.223·20-s − 1.06·22-s − 0.834·23-s + 1/5·25-s − 0.188·28-s − 0.928·29-s + 1.07·31-s − 0.176·32-s + 0.514·34-s − 0.169·35-s + 0.328·37-s − 1.13·38-s − 0.158·40-s + 0.468·41-s − 0.914·43-s + 0.753·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 106470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 106470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + T + p T^{2} \) | 1.47.b |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 11 T + p T^{2} \) | 1.79.l |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 3 T + p T^{2} \) | 1.89.d |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−13.98427733550037, −13.41779537073624, −13.06312464423401, −12.28030336658390, −11.88984881381806, −11.44342089964803, −11.15263121240636, −10.21197910412581, −9.973535917201445, −9.512631461710415, −9.042314844840368, −8.677045934482426, −7.989086429614969, −7.427872370956346, −6.890735664903618, −6.489556614821667, −5.929807154246745, −5.504553793715513, −4.652417841709517, −4.074152565080272, −3.457054558252705, −2.869878087296144, −2.147014223704964, −1.467096274396189, −0.9732272457622756, 0,
0.9732272457622756, 1.467096274396189, 2.147014223704964, 2.869878087296144, 3.457054558252705, 4.074152565080272, 4.652417841709517, 5.504553793715513, 5.929807154246745, 6.489556614821667, 6.890735664903618, 7.427872370956346, 7.989086429614969, 8.677045934482426, 9.042314844840368, 9.512631461710415, 9.973535917201445, 10.21197910412581, 11.15263121240636, 11.44342089964803, 11.88984881381806, 12.28030336658390, 13.06312464423401, 13.41779537073624, 13.98427733550037
