| L(s) = 1 | + (4.89 − 2.82i)2-s + (15.9 − 27.7i)4-s + (−150. − 86.9i)5-s + (242 + 419. i)7-s − 181. i·8-s − 984·10-s + (1.16e3 − 670. i)11-s + (−1.68e3 + 2.91e3i)13-s + (2.37e3 + 1.36e3i)14-s + (−512. − 886. i)16-s + 12.7i·17-s + 5.74e3·19-s + (−4.82e3 + 2.78e3i)20-s + (3.79e3 − 6.56e3i)22-s + (2.92e3 + 1.68e3i)23-s + ⋯ |
| L(s) = 1 | + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (−1.20 − 0.695i)5-s + (0.705 + 1.22i)7-s − 0.353i·8-s − 0.983·10-s + (0.872 − 0.503i)11-s + (−0.766 + 1.32i)13-s + (0.864 + 0.498i)14-s + (−0.125 − 0.216i)16-s + 0.00259i·17-s + 0.837·19-s + (−0.602 + 0.347i)20-s + (0.356 − 0.616i)22-s + (0.240 + 0.138i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.598207004\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.598207004\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-4.89 + 2.82i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (150. + 86.9i)T + (7.81e3 + 1.35e4i)T^{2} \) |
| 7 | \( 1 + (-242 - 419. i)T + (-5.88e4 + 1.01e5i)T^{2} \) |
| 11 | \( 1 + (-1.16e3 + 670. i)T + (8.85e5 - 1.53e6i)T^{2} \) |
| 13 | \( 1 + (1.68e3 - 2.91e3i)T + (-2.41e6 - 4.18e6i)T^{2} \) |
| 17 | \( 1 - 12.7iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 5.74e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + (-2.92e3 - 1.68e3i)T + (7.40e7 + 1.28e8i)T^{2} \) |
| 29 | \( 1 + (-2.54e4 + 1.46e4i)T + (2.97e8 - 5.15e8i)T^{2} \) |
| 31 | \( 1 + (-1.98e4 + 3.44e4i)T + (-4.43e8 - 7.68e8i)T^{2} \) |
| 37 | \( 1 - 5.25e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + (-3.20e4 - 1.85e4i)T + (2.37e9 + 4.11e9i)T^{2} \) |
| 43 | \( 1 + (1.90e3 + 3.29e3i)T + (-3.16e9 + 5.47e9i)T^{2} \) |
| 47 | \( 1 + (-6.65e4 + 3.83e4i)T + (5.38e9 - 9.33e9i)T^{2} \) |
| 53 | \( 1 - 2.38e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + (-2.16e5 - 1.24e5i)T + (2.10e10 + 3.65e10i)T^{2} \) |
| 61 | \( 1 + (6.62e3 + 1.14e4i)T + (-2.57e10 + 4.46e10i)T^{2} \) |
| 67 | \( 1 + (8.44e4 - 1.46e5i)T + (-4.52e10 - 7.83e10i)T^{2} \) |
| 71 | \( 1 + 5.31e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 2.36e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + (-1.75e4 - 3.04e4i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 + (9.50e3 - 5.48e3i)T + (1.63e11 - 2.83e11i)T^{2} \) |
| 89 | \( 1 + 1.29e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + (-1.60e5 - 2.78e5i)T + (-4.16e11 + 7.21e11i)T^{2} \) |
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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
−11.86495595429433786855972211989, −11.33150400763226096692522321342, −9.503173716956654058995307638188, −8.683619810380128394899485499931, −7.58726822478093512114565848737, −6.11885008651945158997091741295, −4.82963678373416413121488419196, −4.07260889787333175308090272135, −2.47834121020123300726526553451, −0.974702470519451748452545281303,
0.857360581248139349664254015392, 3.03117855075388977782798148846, 4.05628146208289659304046691655, 5.01630530401509216395095038427, 6.81732794773635628134895952984, 7.43872737615214771500091752394, 8.220869518635133166171206790474, 10.06249281884288094783441628962, 10.99732356218261132213437846764, 11.83420020351124353544162047054
