| L(s) = 1 | − 3-s − 3·5-s + 9-s − 11-s − 13-s + 3·15-s − 8·17-s + 7·19-s − 2·23-s + 4·25-s − 27-s + 29-s − 4·31-s + 33-s + 3·37-s + 39-s − 6·41-s − 4·43-s − 3·45-s + 47-s + 8·51-s − 6·53-s + 3·55-s − 7·57-s + 3·59-s + 14·61-s + 3·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s + 1/3·9-s − 0.301·11-s − 0.277·13-s + 0.774·15-s − 1.94·17-s + 1.60·19-s − 0.417·23-s + 4/5·25-s − 0.192·27-s + 0.185·29-s − 0.718·31-s + 0.174·33-s + 0.493·37-s + 0.160·39-s − 0.937·41-s − 0.609·43-s − 0.447·45-s + 0.145·47-s + 1.12·51-s − 0.824·53-s + 0.404·55-s − 0.927·57-s + 0.390·59-s + 1.79·61-s + 0.372·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 103488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4862589783\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4862589783\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 + 8 T + p T^{2} \) | 1.17.i |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - T + p T^{2} \) | 1.47.ab |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 - 16 T + p T^{2} \) | 1.97.aq |
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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
−13.55346582779317, −13.21772878240658, −12.68302281377879, −12.10686486287107, −11.70660705048348, −11.33560074943007, −10.99348568978684, −10.42001323479906, −9.723140675879291, −9.388424809755192, −8.613608860041101, −8.217994437776281, −7.661115515220002, −7.208786522126644, −6.702739021888581, −6.281056480427047, −5.252893089092614, −5.143844210082779, −4.432856896075755, −3.848388840394952, −3.467826095170013, −2.606802598838960, −1.993673825430416, −1.029238887467164, −0.2618694448550980,
0.2618694448550980, 1.029238887467164, 1.993673825430416, 2.606802598838960, 3.467826095170013, 3.848388840394952, 4.432856896075755, 5.143844210082779, 5.252893089092614, 6.281056480427047, 6.702739021888581, 7.208786522126644, 7.661115515220002, 8.217994437776281, 8.613608860041101, 9.388424809755192, 9.723140675879291, 10.42001323479906, 10.99348568978684, 11.33560074943007, 11.70660705048348, 12.10686486287107, 12.68302281377879, 13.21772878240658, 13.55346582779317
