| L(s) = 1 | + (−11.3 + 19.5i)2-s + (−304. + 175. i)3-s + (−255. − 443. i)4-s + (−3.22e3 − 1.86e3i)5-s − 7.96e3i·6-s + (4.14e3 + 1.62e4i)7-s + 1.15e4·8-s + (3.23e4 − 5.60e4i)9-s + (7.30e4 − 4.21e4i)10-s + (6.24e4 + 1.08e5i)11-s + (1.56e5 + 9.00e4i)12-s − 3.53e5i·13-s + (−3.66e5 − 1.03e5i)14-s + 1.31e6·15-s + (−1.31e5 + 2.27e5i)16-s + (1.67e6 − 9.67e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.612i)2-s + (−1.25 + 0.724i)3-s + (−0.249 − 0.433i)4-s + (−1.03 − 0.596i)5-s − 1.02i·6-s + (0.246 + 0.969i)7-s + 0.353·8-s + (0.548 − 0.949i)9-s + (0.730 − 0.421i)10-s + (0.387 + 0.671i)11-s + (0.627 + 0.362i)12-s − 0.952i·13-s + (−0.680 − 0.191i)14-s + 1.72·15-s + (−0.125 + 0.216i)16-s + (1.18 − 0.681i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 + 0.390i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.920 + 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.423973 - 0.0861969i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.423973 - 0.0861969i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (11.3 - 19.5i)T \) |
| 7 | \( 1 + (-4.14e3 - 1.62e4i)T \) |
| good | 3 | \( 1 + (304. - 175. i)T + (2.95e4 - 5.11e4i)T^{2} \) |
| 5 | \( 1 + (3.22e3 + 1.86e3i)T + (4.88e6 + 8.45e6i)T^{2} \) |
| 11 | \( 1 + (-6.24e4 - 1.08e5i)T + (-1.29e10 + 2.24e10i)T^{2} \) |
| 13 | \( 1 + 3.53e5iT - 1.37e11T^{2} \) |
| 17 | \( 1 + (-1.67e6 + 9.67e5i)T + (1.00e12 - 1.74e12i)T^{2} \) |
| 19 | \( 1 + (2.53e6 + 1.46e6i)T + (3.06e12 + 5.30e12i)T^{2} \) |
| 23 | \( 1 + (2.94e6 - 5.09e6i)T + (-2.07e13 - 3.58e13i)T^{2} \) |
| 29 | \( 1 + 2.02e7T + 4.20e14T^{2} \) |
| 31 | \( 1 + (-4.61e7 + 2.66e7i)T + (4.09e14 - 7.09e14i)T^{2} \) |
| 37 | \( 1 + (-2.79e7 + 4.84e7i)T + (-2.40e15 - 4.16e15i)T^{2} \) |
| 41 | \( 1 - 8.80e7iT - 1.34e16T^{2} \) |
| 43 | \( 1 - 2.63e8T + 2.16e16T^{2} \) |
| 47 | \( 1 + (2.26e8 + 1.30e8i)T + (2.62e16 + 4.55e16i)T^{2} \) |
| 53 | \( 1 + (1.02e8 + 1.77e8i)T + (-8.74e16 + 1.51e17i)T^{2} \) |
| 59 | \( 1 + (3.01e8 - 1.73e8i)T + (2.55e17 - 4.42e17i)T^{2} \) |
| 61 | \( 1 + (-2.29e8 - 1.32e8i)T + (3.56e17 + 6.17e17i)T^{2} \) |
| 67 | \( 1 + (8.34e8 + 1.44e9i)T + (-9.11e17 + 1.57e18i)T^{2} \) |
| 71 | \( 1 - 1.04e9T + 3.25e18T^{2} \) |
| 73 | \( 1 + (-2.09e9 + 1.20e9i)T + (2.14e18 - 3.72e18i)T^{2} \) |
| 79 | \( 1 + (1.34e8 - 2.32e8i)T + (-4.73e18 - 8.19e18i)T^{2} \) |
| 83 | \( 1 + 2.25e9iT - 1.55e19T^{2} \) |
| 89 | \( 1 + (6.78e9 + 3.91e9i)T + (1.55e19 + 2.70e19i)T^{2} \) |
| 97 | \( 1 + 1.20e10iT - 7.37e19T^{2} \) |
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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
−16.90943776573152123625563770069, −15.80359465327851529631958659590, −15.09141984403996796623217591505, −12.38703560032446528003888611018, −11.35772759394288985913978290303, −9.655183613105332498609478590547, −7.933791055764332210771559957469, −5.78085092774065459885464283846, −4.55425793032650860312453221801, −0.35688558393097108074056757498,
1.09962673794057555234088261618, 3.97592395717945952878778776025, 6.52933100587596248674870681661, 7.927910878419222128364332395055, 10.56912712841523396015155473485, 11.45072596186546825344048231099, 12.44418385725924707623412111609, 14.26437978898123028135537757084, 16.48931263894698616949718217649, 17.27124653000737294315563097411
