| L(s) = 1 | − 11.3i·2-s + (−63 − 50.9i)3-s − 128.·4-s − 576. i·5-s + (−576 + 712. i)6-s + 2.78e3·7-s + 1.44e3i·8-s + (1.37e3 + 6.41e3i)9-s − 6.52e3·10-s − 2.24e4i·11-s + (8.06e3 + 6.51e3i)12-s − 1.31e4·13-s − 3.15e4i·14-s + (−2.93e4 + 3.63e4i)15-s + 1.63e4·16-s + 6.63e4i·17-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + (−0.777 − 0.628i)3-s − 0.500·4-s − 0.923i·5-s + (−0.444 + 0.549i)6-s + 1.16·7-s + 0.353i·8-s + (0.209 + 0.977i)9-s − 0.652·10-s − 1.53i·11-s + (0.388 + 0.314i)12-s − 0.460·13-s − 0.820i·14-s + (−0.580 + 0.718i)15-s + 0.250·16-s + 0.794i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.628 + 0.777i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.628 + 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.442514 - 0.926553i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.442514 - 0.926553i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 11.3iT \) |
| 3 | \( 1 + (63 + 50.9i)T \) |
| good | 5 | \( 1 + 576. iT - 3.90e5T^{2} \) |
| 7 | \( 1 - 2.78e3T + 5.76e6T^{2} \) |
| 11 | \( 1 + 2.24e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 1.31e4T + 8.15e8T^{2} \) |
| 17 | \( 1 - 6.63e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 1.44e5T + 1.69e10T^{2} \) |
| 23 | \( 1 + 4.93e4iT - 7.83e10T^{2} \) |
| 29 | \( 1 - 6.27e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 7.28e5T + 8.52e11T^{2} \) |
| 37 | \( 1 + 1.96e6T + 3.51e12T^{2} \) |
| 41 | \( 1 + 9.86e5iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 7.81e4T + 1.16e13T^{2} \) |
| 47 | \( 1 - 3.51e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 + 5.22e5iT - 6.22e13T^{2} \) |
| 59 | \( 1 - 5.00e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 1.75e7T + 1.91e14T^{2} \) |
| 67 | \( 1 + 1.71e7T + 4.06e14T^{2} \) |
| 71 | \( 1 + 2.58e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 2.81e7T + 8.06e14T^{2} \) |
| 79 | \( 1 - 9.18e6T + 1.51e15T^{2} \) |
| 83 | \( 1 - 8.71e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 8.12e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 1.28e8T + 7.83e15T^{2} \) |
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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
−20.89584655302333174624205760107, −19.27614328231768897055248638516, −17.83556206177773251820734599158, −16.58171257300205966983593566690, −13.83286907179083159349749942332, −12.26671734216539629219257399060, −10.99764242447386960739802118707, −8.338277152980955213364917201137, −5.20770109877216866850706186223, −1.11307400118176072275291948040,
4.89825008444268207342800220819, 7.17854412970097166907759560186, 9.952645771016127759046600591714, 11.73615780025091426231820046210, 14.45977038491012259071190772770, 15.50932813861659101071182746475, 17.38711976545548378528274612167, 18.18511240999852865890420946834, 20.71922738888534525984863229335, 22.29562385070242310283253082095
