| L(s) = 1 | + (−13.5 − 23.4i)2-s + (−121.5 − 70.1i)3-s + (146. − 253. i)4-s + (−1.21e3 + 699. i)5-s + 3.79e3i·6-s + (−1.29e4 − 1.07e4i)7-s − 3.56e4·8-s + (9.84e3 + 1.70e4i)9-s + (3.27e4 + 1.89e4i)10-s + (−3.96e3 + 6.87e3i)11-s + (−3.55e4 + 2.05e4i)12-s + 8.08e4i·13-s + (−7.60e4 + 4.48e5i)14-s + 1.96e5·15-s + (3.31e5 + 5.74e5i)16-s + (1.93e6 + 1.11e6i)17-s + ⋯ |
| L(s) = 1 | + (−0.422 − 0.732i)2-s + (−0.5 − 0.288i)3-s + (0.142 − 0.247i)4-s + (−0.387 + 0.223i)5-s + 0.488i·6-s + (−0.770 − 0.637i)7-s − 1.08·8-s + (0.166 + 0.288i)9-s + (0.327 + 0.189i)10-s + (−0.0246 + 0.0426i)11-s + (−0.142 + 0.0824i)12-s + 0.217i·13-s + (−0.141 + 0.833i)14-s + 0.258·15-s + (0.316 + 0.548i)16-s + (1.36 + 0.788i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.535 - 0.844i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.238093 + 0.130990i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.238093 + 0.130990i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (121.5 + 70.1i)T \) |
| 7 | \( 1 + (1.29e4 + 1.07e4i)T \) |
| good | 2 | \( 1 + (13.5 + 23.4i)T + (-512 + 886. i)T^{2} \) |
| 5 | \( 1 + (1.21e3 - 699. i)T + (4.88e6 - 8.45e6i)T^{2} \) |
| 11 | \( 1 + (3.96e3 - 6.87e3i)T + (-1.29e10 - 2.24e10i)T^{2} \) |
| 13 | \( 1 - 8.08e4iT - 1.37e11T^{2} \) |
| 17 | \( 1 + (-1.93e6 - 1.11e6i)T + (1.00e12 + 1.74e12i)T^{2} \) |
| 19 | \( 1 + (-5.41e5 + 3.12e5i)T + (3.06e12 - 5.30e12i)T^{2} \) |
| 23 | \( 1 + (-2.73e6 - 4.73e6i)T + (-2.07e13 + 3.58e13i)T^{2} \) |
| 29 | \( 1 - 2.70e6T + 4.20e14T^{2} \) |
| 31 | \( 1 + (3.76e7 + 2.17e7i)T + (4.09e14 + 7.09e14i)T^{2} \) |
| 37 | \( 1 + (1.77e7 + 3.07e7i)T + (-2.40e15 + 4.16e15i)T^{2} \) |
| 41 | \( 1 - 4.22e7iT - 1.34e16T^{2} \) |
| 43 | \( 1 + 7.33e7T + 2.16e16T^{2} \) |
| 47 | \( 1 + (1.77e8 - 1.02e8i)T + (2.62e16 - 4.55e16i)T^{2} \) |
| 53 | \( 1 + (3.86e8 - 6.69e8i)T + (-8.74e16 - 1.51e17i)T^{2} \) |
| 59 | \( 1 + (-2.58e8 - 1.49e8i)T + (2.55e17 + 4.42e17i)T^{2} \) |
| 61 | \( 1 + (-1.13e9 + 6.53e8i)T + (3.56e17 - 6.17e17i)T^{2} \) |
| 67 | \( 1 + (6.76e8 - 1.17e9i)T + (-9.11e17 - 1.57e18i)T^{2} \) |
| 71 | \( 1 + 1.58e9T + 3.25e18T^{2} \) |
| 73 | \( 1 + (3.09e9 + 1.78e9i)T + (2.14e18 + 3.72e18i)T^{2} \) |
| 79 | \( 1 + (1.98e9 + 3.43e9i)T + (-4.73e18 + 8.19e18i)T^{2} \) |
| 83 | \( 1 - 1.06e9iT - 1.55e19T^{2} \) |
| 89 | \( 1 + (-4.10e9 + 2.36e9i)T + (1.55e19 - 2.70e19i)T^{2} \) |
| 97 | \( 1 - 9.72e9iT - 7.37e19T^{2} \) |
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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
−16.13994643701422093880144186675, −14.73541274378327627150169479313, −13.00607338958247510866300317635, −11.74832177323097440399669863417, −10.65773278772574533551535327952, −9.544185692422151905480251737161, −7.39170025026557462807455562585, −5.87914914727177550263092731284, −3.41005356695702276633020747770, −1.34087487266107092705038512838,
0.14816563944462636425699325465, 3.23091951910956032678074036842, 5.51321192904432493521508487236, 6.93981424836505080244839980045, 8.451755160414230997481654180076, 9.830288917699230905656239782831, 11.74730916118748728087195393432, 12.62792703707336217886915781834, 14.77154691991675661205423809406, 16.09081406548986554076866605222
