| L(s) = 1 | + (4.89 + 2.82i)2-s + (15.9 + 27.7i)4-s + (39.5 − 22.8i)5-s + (−245. + 424. i)7-s + 181. i·8-s + 258.·10-s + (873. + 504. i)11-s + (466. + 808. i)13-s + (−2.40e3 + 1.38e3i)14-s + (−512. + 886. i)16-s + 8.09e3i·17-s − 7.72e3·19-s + (1.26e3 + 730. i)20-s + (2.85e3 + 4.94e3i)22-s + (1.18e4 − 6.84e3i)23-s + ⋯ |
| L(s) = 1 | + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (0.316 − 0.182i)5-s + (−0.714 + 1.23i)7-s + 0.353i·8-s + 0.258·10-s + (0.656 + 0.378i)11-s + (0.212 + 0.368i)13-s + (−0.875 + 0.505i)14-s + (−0.125 + 0.216i)16-s + 1.64i·17-s − 1.12·19-s + (0.158 + 0.0913i)20-s + (0.267 + 0.463i)22-s + (0.973 − 0.562i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.250 - 0.968i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.250 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.38706 + 1.79153i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.38706 + 1.79153i\) |
| \(L(4)\) |
|
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 | \( 1 + (-39.5 + 22.8i)T + (7.81e3 - 1.35e4i)T^{2} \) |
| 7 | \( 1 + (245. - 424. i)T + (-5.88e4 - 1.01e5i)T^{2} \) |
| 11 | \( 1 + (-873. - 504. i)T + (8.85e5 + 1.53e6i)T^{2} \) |
| 13 | \( 1 + (-466. - 808. i)T + (-2.41e6 + 4.18e6i)T^{2} \) |
| 17 | \( 1 - 8.09e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 7.72e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + (-1.18e4 + 6.84e3i)T + (7.40e7 - 1.28e8i)T^{2} \) |
| 29 | \( 1 + (-1.96e3 - 1.13e3i)T + (2.97e8 + 5.15e8i)T^{2} \) |
| 31 | \( 1 + (1.70e4 + 2.95e4i)T + (-4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 - 9.20e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + (-3.10e4 + 1.79e4i)T + (2.37e9 - 4.11e9i)T^{2} \) |
| 43 | \( 1 + (-3.45e4 + 5.98e4i)T + (-3.16e9 - 5.47e9i)T^{2} \) |
| 47 | \( 1 + (1.32e4 + 7.62e3i)T + (5.38e9 + 9.33e9i)T^{2} \) |
| 53 | \( 1 + 2.36e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + (-2.21e5 + 1.28e5i)T + (2.10e10 - 3.65e10i)T^{2} \) |
| 61 | \( 1 + (1.99e4 - 3.45e4i)T + (-2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (1.60e5 + 2.77e5i)T + (-4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 - 4.04e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 3.93e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + (4.49e5 - 7.78e5i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 + (1.54e5 + 8.94e4i)T + (1.63e11 + 2.83e11i)T^{2} \) |
| 89 | \( 1 - 8.26e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + (3.17e5 - 5.50e5i)T + (-4.16e11 - 7.21e11i)T^{2} \) |
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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
−14.71421162865111464301597915285, −13.04471205880263538564438687407, −12.56462219994293186400320034913, −11.20083561278081733991352048567, −9.503794125947215072596538914525, −8.468669117465410446609428694283, −6.59893012967823543134982892963, −5.72072481346119925741695671204, −3.98833667757479110446945930866, −2.17032737102348169732812701438,
0.820851924587811420712100752356, 2.98890384039902888041420914703, 4.37689878444401565362220546563, 6.15438560297071371165813768455, 7.27298058553512326510452311590, 9.291707055739653693284856296690, 10.42016089154208702987336861924, 11.42391979402481627753984866887, 12.88630525145678174102640103612, 13.68143289047306500716177704260
