| L(s) = 1 | − 3-s − 5-s − 7-s + 9-s + 3·11-s + 15-s − 5·19-s + 21-s − 6·23-s + 25-s − 27-s − 6·29-s + 7·31-s − 3·33-s + 35-s − 4·37-s + 3·41-s + 4·43-s − 45-s + 3·47-s + 49-s + 6·53-s − 3·55-s + 5·57-s − 6·59-s + 14·61-s − 63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.904·11-s + 0.258·15-s − 1.14·19-s + 0.218·21-s − 1.25·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s + 1.25·31-s − 0.522·33-s + 0.169·35-s − 0.657·37-s + 0.468·41-s + 0.609·43-s − 0.149·45-s + 0.437·47-s + 1/7·49-s + 0.824·53-s − 0.404·55-s + 0.662·57-s − 0.781·59-s + 1.79·61-s − 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 283920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.696672006\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.696672006\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 - 5 T + p T^{2} \) | 1.97.af |
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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
−12.55748130477886, −12.15438720027562, −11.94607185446345, −11.43407944555397, −10.84500553509620, −10.58412296068218, −9.990902834023495, −9.566603703270877, −9.049583837712711, −8.600130428161880, −8.026708599316507, −7.645730322606384, −6.918398159367201, −6.671719070157991, −6.104976720366060, −5.771488372472060, −5.115980585283649, −4.474369403274940, −4.010776099322005, −3.758190620827386, −3.035949004436324, −2.166755536492052, −1.895359120159944, −0.8755835847894213, −0.4580316313504197,
0.4580316313504197, 0.8755835847894213, 1.895359120159944, 2.166755536492052, 3.035949004436324, 3.758190620827386, 4.010776099322005, 4.474369403274940, 5.115980585283649, 5.771488372472060, 6.104976720366060, 6.671719070157991, 6.918398159367201, 7.645730322606384, 8.026708599316507, 8.600130428161880, 9.049583837712711, 9.566603703270877, 9.990902834023495, 10.58412296068218, 10.84500553509620, 11.43407944555397, 11.94607185446345, 12.15438720027562, 12.55748130477886
