| L(s) = 1 | + 3-s + 9-s + 6·11-s + 4·13-s + 6·17-s − 2·19-s + 27-s − 6·29-s − 10·31-s + 6·33-s + 2·37-s + 4·39-s + 6·41-s + 4·43-s + 6·51-s − 12·53-s − 2·57-s + 14·61-s + 4·67-s − 6·71-s − 4·73-s + 16·79-s + 81-s − 12·83-s − 6·87-s − 6·89-s − 10·93-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s + 1.80·11-s + 1.10·13-s + 1.45·17-s − 0.458·19-s + 0.192·27-s − 1.11·29-s − 1.79·31-s + 1.04·33-s + 0.328·37-s + 0.640·39-s + 0.937·41-s + 0.609·43-s + 0.840·51-s − 1.64·53-s − 0.264·57-s + 1.79·61-s + 0.488·67-s − 0.712·71-s − 0.468·73-s + 1.80·79-s + 1/9·81-s − 1.31·83-s − 0.643·87-s − 0.635·89-s − 1.03·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.957199501\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.957199501\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 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 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 16 T + p T^{2} \) | 1.97.q |
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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
−12.95937482530613, −12.42102453978920, −12.13311228724065, −11.36633512480722, −11.10495204014170, −10.76277470615956, −9.876551104071068, −9.536153482605101, −9.288806585215673, −8.578965001345556, −8.433709036971953, −7.642753469176245, −7.293262152545195, −6.806603797571619, −6.036282759179750, −5.940671223553891, −5.262878213774854, −4.457079444404971, −3.922943611908806, −3.615530579456898, −3.243387986253208, −2.347555773196730, −1.659367600102264, −1.320590092455971, −0.6149878599005496,
0.6149878599005496, 1.320590092455971, 1.659367600102264, 2.347555773196730, 3.243387986253208, 3.615530579456898, 3.922943611908806, 4.457079444404971, 5.262878213774854, 5.940671223553891, 6.036282759179750, 6.806603797571619, 7.293262152545195, 7.642753469176245, 8.433709036971953, 8.578965001345556, 9.288806585215673, 9.536153482605101, 9.876551104071068, 10.76277470615956, 11.10495204014170, 11.36633512480722, 12.13311228724065, 12.42102453978920, 12.95937482530613
