| L(s) = 1 | − 3-s − 2·4-s − 3·5-s − 2·9-s − 11-s + 2·12-s + 4·13-s + 3·15-s + 4·16-s + 6·17-s − 2·19-s + 6·20-s + 3·23-s + 4·25-s + 5·27-s − 6·29-s − 5·31-s + 33-s + 4·36-s + 11·37-s − 4·39-s − 6·41-s + 8·43-s + 2·44-s + 6·45-s − 4·48-s − 6·51-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 1.34·5-s − 2/3·9-s − 0.301·11-s + 0.577·12-s + 1.10·13-s + 0.774·15-s + 16-s + 1.45·17-s − 0.458·19-s + 1.34·20-s + 0.625·23-s + 4/5·25-s + 0.962·27-s − 1.11·29-s − 0.898·31-s + 0.174·33-s + 2/3·36-s + 1.80·37-s − 0.640·39-s − 0.937·41-s + 1.21·43-s + 0.301·44-s + 0.894·45-s − 0.577·48-s − 0.840·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5869444112\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5869444112\) |
| \(L(\frac{3}{2})\) |
|
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 | 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 - 11 T + p T^{2} \) | 1.37.al |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 9 T + p T^{2} \) | 1.59.aj |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 97 | \( 1 - T + p T^{2} \) | 1.97.ab |
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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
−11.10866631247395777390781457672, −9.985286197483592787716573688901, −8.868335617542485650901555955819, −8.196526304421111242297150106130, −7.44927930218322722277526119885, −5.98016833623442124300848518324, −5.22586837194321912962810216772, −4.06364144920564711800863579864, −3.31516413382692155506861780875, −0.70813066257818517183339634810,
0.70813066257818517183339634810, 3.31516413382692155506861780875, 4.06364144920564711800863579864, 5.22586837194321912962810216772, 5.98016833623442124300848518324, 7.44927930218322722277526119885, 8.196526304421111242297150106130, 8.868335617542485650901555955819, 9.985286197483592787716573688901, 11.10866631247395777390781457672
