| L(s) = 1 | + (4.89 + 2.82i)2-s + (29.9 + 17.2i)3-s + (15.9 + 27.7i)4-s + (48.9 − 84.8i)5-s + (97.7 + 169. i)6-s + 374.·7-s + 181. i·8-s + (232. + 403. i)9-s + (480. − 277. i)10-s − 766.·11-s + 1.10e3i·12-s + (−1.99e3 + 1.15e3i)13-s + (1.83e3 + 1.05e3i)14-s + (2.93e3 − 1.69e3i)15-s + (−512. + 886. i)16-s + (1.07e3 − 1.86e3i)17-s + ⋯ |
| L(s) = 1 | + (0.612 + 0.353i)2-s + (1.10 + 0.640i)3-s + (0.249 + 0.433i)4-s + (0.391 − 0.678i)5-s + (0.452 + 0.783i)6-s + 1.09·7-s + 0.353i·8-s + (0.319 + 0.553i)9-s + (0.480 − 0.277i)10-s − 0.575·11-s + 0.640i·12-s + (−0.910 + 0.525i)13-s + (0.668 + 0.385i)14-s + (0.869 − 0.501i)15-s + (−0.125 + 0.216i)16-s + (0.219 − 0.380i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.658 - 0.752i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.658 - 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(3.21060 + 1.45564i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.21060 + 1.45564i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-4.89 - 2.82i)T \) |
| 19 | \( 1 + (-1.56e3 - 6.67e3i)T \) |
| good | 3 | \( 1 + (-29.9 - 17.2i)T + (364.5 + 631. i)T^{2} \) |
| 5 | \( 1 + (-48.9 + 84.8i)T + (-7.81e3 - 1.35e4i)T^{2} \) |
| 7 | \( 1 - 374.T + 1.17e5T^{2} \) |
| 11 | \( 1 + 766.T + 1.77e6T^{2} \) |
| 13 | \( 1 + (1.99e3 - 1.15e3i)T + (2.41e6 - 4.18e6i)T^{2} \) |
| 17 | \( 1 + (-1.07e3 + 1.86e3i)T + (-1.20e7 - 2.09e7i)T^{2} \) |
| 23 | \( 1 + (7.67e3 + 1.32e4i)T + (-7.40e7 + 1.28e8i)T^{2} \) |
| 29 | \( 1 + (1.65e4 - 9.53e3i)T + (2.97e8 - 5.15e8i)T^{2} \) |
| 31 | \( 1 + 4.99e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + 3.25e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + (3.58e4 + 2.06e4i)T + (2.37e9 + 4.11e9i)T^{2} \) |
| 43 | \( 1 + (1.13e4 - 1.95e4i)T + (-3.16e9 - 5.47e9i)T^{2} \) |
| 47 | \( 1 + (1.46e4 + 2.54e4i)T + (-5.38e9 + 9.33e9i)T^{2} \) |
| 53 | \( 1 + (-4.36e4 + 2.52e4i)T + (1.10e10 - 1.91e10i)T^{2} \) |
| 59 | \( 1 + (-2.68e5 - 1.55e5i)T + (2.10e10 + 3.65e10i)T^{2} \) |
| 61 | \( 1 + (-1.50e5 - 2.61e5i)T + (-2.57e10 + 4.46e10i)T^{2} \) |
| 67 | \( 1 + (13.3 - 7.70i)T + (4.52e10 - 7.83e10i)T^{2} \) |
| 71 | \( 1 + (-3.15e5 - 1.82e5i)T + (6.40e10 + 1.10e11i)T^{2} \) |
| 73 | \( 1 + (6.10e4 - 1.05e5i)T + (-7.56e10 - 1.31e11i)T^{2} \) |
| 79 | \( 1 + (6.44e5 + 3.71e5i)T + (1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 + 1.00e6T + 3.26e11T^{2} \) |
| 89 | \( 1 + (-1.66e5 + 9.63e4i)T + (2.48e11 - 4.30e11i)T^{2} \) |
| 97 | \( 1 + (-1.37e6 - 7.95e5i)T + (4.16e11 + 7.21e11i)T^{2} \) |
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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
−14.75509914156291608574787433681, −14.39095110944941648265399095964, −13.13703408998880218784667301697, −11.76763299284450268975368287168, −9.985083234586467684946076790200, −8.714048147956349191016476012558, −7.66490362458587391922066306140, −5.36299585715767632681148527692, −4.16280282031106688076181677884, −2.25454233562225832459568722139,
1.86488723470750523902349089654, 3.01303481610322638773998526851, 5.17063457437932088409383338906, 7.14614377177390291189585953656, 8.260076854431244430521435632472, 10.01516950658872019004970327195, 11.31499215915150179621982044106, 12.76623178162506274582606568966, 13.85591346884665207265790205587, 14.49294917284954313757854574668
