| L(s) = 1 | + (35.5 + 20.4i)3-s + (20.1 − 34.9i)5-s + 53.8·7-s + (475. + 824. i)9-s + 1.99e3·11-s + (1.39e3 − 805. i)13-s + (1.43e3 − 826. i)15-s + (−1.59e3 + 2.75e3i)17-s + (−2.91e3 + 6.20e3i)19-s + (1.91e3 + 1.10e3i)21-s + (−3.37e3 − 5.84e3i)23-s + (6.99e3 + 1.21e4i)25-s + 9.14e3i·27-s + (454. − 262. i)29-s − 1.00e4i·31-s + ⋯ |
| L(s) = 1 | + (1.31 + 0.759i)3-s + (0.161 − 0.279i)5-s + 0.156·7-s + (0.652 + 1.13i)9-s + 1.49·11-s + (0.634 − 0.366i)13-s + (0.424 − 0.244i)15-s + (−0.323 + 0.560i)17-s + (−0.424 + 0.905i)19-s + (0.206 + 0.119i)21-s + (−0.277 − 0.480i)23-s + (0.447 + 0.775i)25-s + 0.464i·27-s + (0.0186 − 0.0107i)29-s − 0.338i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.760 - 0.649i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.760 - 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(3.04278 + 1.12192i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.04278 + 1.12192i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (2.91e3 - 6.20e3i)T \) |
| good | 3 | \( 1 + (-35.5 - 20.4i)T + (364.5 + 631. i)T^{2} \) |
| 5 | \( 1 + (-20.1 + 34.9i)T + (-7.81e3 - 1.35e4i)T^{2} \) |
| 7 | \( 1 - 53.8T + 1.17e5T^{2} \) |
| 11 | \( 1 - 1.99e3T + 1.77e6T^{2} \) |
| 13 | \( 1 + (-1.39e3 + 805. i)T + (2.41e6 - 4.18e6i)T^{2} \) |
| 17 | \( 1 + (1.59e3 - 2.75e3i)T + (-1.20e7 - 2.09e7i)T^{2} \) |
| 23 | \( 1 + (3.37e3 + 5.84e3i)T + (-7.40e7 + 1.28e8i)T^{2} \) |
| 29 | \( 1 + (-454. + 262. i)T + (2.97e8 - 5.15e8i)T^{2} \) |
| 31 | \( 1 + 1.00e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 - 4.50e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + (-6.67e4 - 3.85e4i)T + (2.37e9 + 4.11e9i)T^{2} \) |
| 43 | \( 1 + (-2.47e4 + 4.28e4i)T + (-3.16e9 - 5.47e9i)T^{2} \) |
| 47 | \( 1 + (4.69e3 + 8.12e3i)T + (-5.38e9 + 9.33e9i)T^{2} \) |
| 53 | \( 1 + (-5.47e4 + 3.16e4i)T + (1.10e10 - 1.91e10i)T^{2} \) |
| 59 | \( 1 + (2.41e5 + 1.39e5i)T + (2.10e10 + 3.65e10i)T^{2} \) |
| 61 | \( 1 + (2.11e5 + 3.66e5i)T + (-2.57e10 + 4.46e10i)T^{2} \) |
| 67 | \( 1 + (2.97e3 - 1.71e3i)T + (4.52e10 - 7.83e10i)T^{2} \) |
| 71 | \( 1 + (1.73e5 + 1.00e5i)T + (6.40e10 + 1.10e11i)T^{2} \) |
| 73 | \( 1 + (-1.20e5 + 2.09e5i)T + (-7.56e10 - 1.31e11i)T^{2} \) |
| 79 | \( 1 + (-5.82e4 - 3.36e4i)T + (1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 + 3.36e5T + 3.26e11T^{2} \) |
| 89 | \( 1 + (-3.45e5 + 1.99e5i)T + (2.48e11 - 4.30e11i)T^{2} \) |
| 97 | \( 1 + (1.20e6 + 6.94e5i)T + (4.16e11 + 7.21e11i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.71054365979141787090569769271, −12.49666802867808681960098618528, −11.01111999886383949301504270278, −9.759455536404673773033785205307, −8.896490199259286778893556003747, −8.067647533333181164111983519705, −6.26981178663510624212607445291, −4.38075472108162183269306265637, −3.39260068835884615479367399875, −1.63025465822378329672620634505,
1.32106620960967366059075378720, 2.65324070840247285361572417156, 4.10178113932209818602332943835, 6.39261275046519201592580060126, 7.34251511846216331961587237822, 8.707048036977598429386882675133, 9.316539588840160049364245580487, 11.05473079735629332268617585478, 12.26121258594730807213810647582, 13.48187109013669962079195454527
