| L(s) = 1 | + (9.23 + 5.33i)2-s + (−1.86 − 26.9i)3-s + (24.9 + 43.1i)4-s + (152. + 88.2i)5-s + (126. − 258. i)6-s + (342. − 17.4i)7-s − 151. i·8-s + (−722. + 100. i)9-s + (941. + 1.63e3i)10-s + (−766. + 442. i)11-s + (1.11e3 − 751. i)12-s − 3.03e3·13-s + (3.25e3 + 1.66e3i)14-s + (2.09e3 − 4.28e3i)15-s + (2.40e3 − 4.15e3i)16-s + (−408. + 235. i)17-s + ⋯ |
| L(s) = 1 | + (1.15 + 0.666i)2-s + (−0.0690 − 0.997i)3-s + (0.389 + 0.674i)4-s + (1.22 + 0.705i)5-s + (0.585 − 1.19i)6-s + (0.998 − 0.0508i)7-s − 0.295i·8-s + (−0.990 + 0.137i)9-s + (0.941 + 1.63i)10-s + (−0.575 + 0.332i)11-s + (0.645 − 0.434i)12-s − 1.38·13-s + (1.18 + 0.607i)14-s + (0.619 − 1.26i)15-s + (0.586 − 1.01i)16-s + (−0.0831 + 0.0480i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.182i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.983 - 0.182i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.90854 + 0.267356i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.90854 + 0.267356i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.86 + 26.9i)T \) |
| 7 | \( 1 + (-342. + 17.4i)T \) |
| good | 2 | \( 1 + (-9.23 - 5.33i)T + (32 + 55.4i)T^{2} \) |
| 5 | \( 1 + (-152. - 88.2i)T + (7.81e3 + 1.35e4i)T^{2} \) |
| 11 | \( 1 + (766. - 442. i)T + (8.85e5 - 1.53e6i)T^{2} \) |
| 13 | \( 1 + 3.03e3T + 4.82e6T^{2} \) |
| 17 | \( 1 + (408. - 235. i)T + (1.20e7 - 2.09e7i)T^{2} \) |
| 19 | \( 1 + (4.20e3 - 7.28e3i)T + (-2.35e7 - 4.07e7i)T^{2} \) |
| 23 | \( 1 + (-489. - 282. i)T + (7.40e7 + 1.28e8i)T^{2} \) |
| 29 | \( 1 + 3.50e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + (-1.29e4 - 2.24e4i)T + (-4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 + (-1.75e3 + 3.03e3i)T + (-1.28e9 - 2.22e9i)T^{2} \) |
| 41 | \( 1 + 1.16e3iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 9.18e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + (-4.77e4 - 2.75e4i)T + (5.38e9 + 9.33e9i)T^{2} \) |
| 53 | \( 1 + (1.48e4 - 8.54e3i)T + (1.10e10 - 1.91e10i)T^{2} \) |
| 59 | \( 1 + (-3.27e5 + 1.89e5i)T + (2.10e10 - 3.65e10i)T^{2} \) |
| 61 | \( 1 + (3.94e4 - 6.83e4i)T + (-2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (-3.01e4 - 5.22e4i)T + (-4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 + 3.30e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + (1.82e5 + 3.15e5i)T + (-7.56e10 + 1.31e11i)T^{2} \) |
| 79 | \( 1 + (-1.83e5 + 3.18e5i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 - 5.18e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + (-6.51e5 - 3.76e5i)T + (2.48e11 + 4.30e11i)T^{2} \) |
| 97 | \( 1 - 1.02e6T + 8.32e11T^{2} \) |
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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
−17.04631642878282773298069940279, −14.90664018426338503252851981712, −14.26451321059169798614956953238, −13.32825633567717944126714763463, −12.09851166163015531751839588051, −10.18029544904293048575862678801, −7.68450827601398360056610129633, −6.38748155048995598292549351078, −5.15585289420854804657798332416, −2.23182380067627805206828699315,
2.41904027653665549078604716311, 4.71808976112878608413184312561, 5.39275624570949748525583273657, 8.733351521319782396951929460318, 10.27025523323651698106284603649, 11.53664851417123597436370584200, 12.97107106413603993996433129597, 14.11054661049431674919637136163, 15.05545511198078127757864635083, 16.89172156097558364452126842605
