| L(s) = 1 | + (−13.1 + 7.60i)2-s + (23.2 + 13.8i)3-s + (83.5 − 144. i)4-s + (82.1 − 47.4i)5-s + (−410. − 5.43i)6-s + (334. − 73.9i)7-s + 1.56e3i·8-s + (347. + 640. i)9-s + (−721. + 1.24e3i)10-s + (−735. − 424. i)11-s + (3.93e3 − 2.20e3i)12-s + 1.14e3·13-s + (−3.84e3 + 3.51e3i)14-s + (2.56e3 + 33.9i)15-s + (−6.56e3 − 1.13e4i)16-s + (2.38e3 + 1.37e3i)17-s + ⋯ |
| L(s) = 1 | + (−1.64 + 0.950i)2-s + (0.859 + 0.511i)3-s + (1.30 − 2.26i)4-s + (0.657 − 0.379i)5-s + (−1.90 − 0.0251i)6-s + (0.976 − 0.215i)7-s + 3.06i·8-s + (0.476 + 0.878i)9-s + (−0.721 + 1.24i)10-s + (−0.552 − 0.319i)11-s + (2.27 − 1.27i)12-s + 0.520·13-s + (−1.40 + 1.28i)14-s + (0.758 + 0.0100i)15-s + (−1.60 − 2.77i)16-s + (0.484 + 0.279i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.373 - 0.927i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.373 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.935912 + 0.631964i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.935912 + 0.631964i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-23.2 - 13.8i)T \) |
| 7 | \( 1 + (-334. + 73.9i)T \) |
| good | 2 | \( 1 + (13.1 - 7.60i)T + (32 - 55.4i)T^{2} \) |
| 5 | \( 1 + (-82.1 + 47.4i)T + (7.81e3 - 1.35e4i)T^{2} \) |
| 11 | \( 1 + (735. + 424. i)T + (8.85e5 + 1.53e6i)T^{2} \) |
| 13 | \( 1 - 1.14e3T + 4.82e6T^{2} \) |
| 17 | \( 1 + (-2.38e3 - 1.37e3i)T + (1.20e7 + 2.09e7i)T^{2} \) |
| 19 | \( 1 + (-3.55e3 - 6.15e3i)T + (-2.35e7 + 4.07e7i)T^{2} \) |
| 23 | \( 1 + (-4.81e3 + 2.78e3i)T + (7.40e7 - 1.28e8i)T^{2} \) |
| 29 | \( 1 + 2.27e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + (4.26e3 - 7.37e3i)T + (-4.43e8 - 7.68e8i)T^{2} \) |
| 37 | \( 1 + (3.59e4 + 6.22e4i)T + (-1.28e9 + 2.22e9i)T^{2} \) |
| 41 | \( 1 - 5.90e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 3.77e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + (-9.76e4 + 5.63e4i)T + (5.38e9 - 9.33e9i)T^{2} \) |
| 53 | \( 1 + (1.75e5 + 1.01e5i)T + (1.10e10 + 1.91e10i)T^{2} \) |
| 59 | \( 1 + (1.50e5 + 8.66e4i)T + (2.10e10 + 3.65e10i)T^{2} \) |
| 61 | \( 1 + (-2.45e4 - 4.25e4i)T + (-2.57e10 + 4.46e10i)T^{2} \) |
| 67 | \( 1 + (1.56e5 - 2.71e5i)T + (-4.52e10 - 7.83e10i)T^{2} \) |
| 71 | \( 1 + 1.00e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + (-1.41e5 + 2.45e5i)T + (-7.56e10 - 1.31e11i)T^{2} \) |
| 79 | \( 1 + (1.86e5 + 3.22e5i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 + 2.43e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + (8.90e5 - 5.13e5i)T + (2.48e11 - 4.30e11i)T^{2} \) |
| 97 | \( 1 + 1.64e6T + 8.32e11T^{2} \) |
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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
−17.03592261317672452716290985931, −16.05933541863906923106406151035, −14.89208196460866681930201077001, −13.84098040430643063987099390068, −10.86244692006124214348344129303, −9.791060271284212179928737096704, −8.586549629890633129928146901397, −7.68542594600736948424775055096, −5.50560610249403930825244547942, −1.60430911438549170470735055525,
1.44711939470708492155529660457, 2.79759300981681913744594043851, 7.23011027414968293634761915983, 8.410389632320798754948348310464, 9.557546039927895182092656048816, 10.87447347902180192681182465874, 12.26105457327831990733538314145, 13.77381547563652372237605799867, 15.52869293016115452652835697656, 17.36814417288500383524786133912
