| L(s) = 1 | + 190.·5-s + 4.09e3i·7-s + 8.80e3i·11-s − 2.75e3·13-s − 9.64e4·17-s − 5.16e3i·19-s + 4.82e5i·23-s − 3.54e5·25-s − 1.15e6·29-s − 4.21e5i·31-s + 7.80e5i·35-s + 2.09e6·37-s + 1.11e6·41-s − 1.00e6i·43-s − 4.11e6i·47-s + ⋯ |
| L(s) = 1 | + 0.304·5-s + 1.70i·7-s + 0.601i·11-s − 0.0963·13-s − 1.15·17-s − 0.0395i·19-s + 1.72i·23-s − 0.907·25-s − 1.63·29-s − 0.456i·31-s + 0.520i·35-s + 1.11·37-s + 0.396·41-s − 0.292i·43-s − 0.842i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.3694687938\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3694687938\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 190.T + 3.90e5T^{2} \) |
| 7 | \( 1 - 4.09e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 - 8.80e3iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 2.75e3T + 8.15e8T^{2} \) |
| 17 | \( 1 + 9.64e4T + 6.97e9T^{2} \) |
| 19 | \( 1 + 5.16e3iT - 1.69e10T^{2} \) |
| 23 | \( 1 - 4.82e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 1.15e6T + 5.00e11T^{2} \) |
| 31 | \( 1 + 4.21e5iT - 8.52e11T^{2} \) |
| 37 | \( 1 - 2.09e6T + 3.51e12T^{2} \) |
| 41 | \( 1 - 1.11e6T + 7.98e12T^{2} \) |
| 43 | \( 1 + 1.00e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + 4.11e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 8.86e6T + 6.22e13T^{2} \) |
| 59 | \( 1 + 1.72e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 1.99e7T + 1.91e14T^{2} \) |
| 67 | \( 1 - 6.64e6iT - 4.06e14T^{2} \) |
| 71 | \( 1 - 1.61e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 2.88e6T + 8.06e14T^{2} \) |
| 79 | \( 1 + 4.80e6iT - 1.51e15T^{2} \) |
| 83 | \( 1 - 2.15e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + 8.89e7T + 3.93e15T^{2} \) |
| 97 | \( 1 - 8.89e7T + 7.83e15T^{2} \) |
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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
−11.25966264497629297405339380912, −9.695175302812170052735561901137, −9.294915479948522843559717178795, −8.249530882905910822798157131295, −7.11996116103432388814544720112, −5.88694047487533508899256577944, −5.30197377789616428254873467674, −3.90509292331087347797170848875, −2.43747695332861021884932182172, −1.82468908796494252273435696501,
0.079355621834174737384007667934, 1.01734971627839334369439187868, 2.37949538473380197196769319295, 3.81009747481828526646276364598, 4.54040319696513960167029464693, 5.98290477669341655295888568088, 6.91346625050162194541076711082, 7.79042273437970457361873198404, 8.887126225968194204382776688269, 9.961242559563353531285496438055
