| L(s) = 1 | + (4.89 − 2.82i)2-s + (15.9 − 27.7i)4-s + (132. + 76.7i)5-s + (−336. − 582. i)7-s − 181. i·8-s + 868.·10-s + (−2.21e3 + 1.27e3i)11-s + (885. − 1.53e3i)13-s + (−3.29e3 − 1.90e3i)14-s + (−512. − 886. i)16-s + 3.97e3i·17-s − 7.76e3·19-s + (4.25e3 − 2.45e3i)20-s + (−7.24e3 + 1.25e4i)22-s + (−2.26e3 − 1.30e3i)23-s + ⋯ |
| L(s) = 1 | + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (1.06 + 0.614i)5-s + (−0.979 − 1.69i)7-s − 0.353i·8-s + 0.868·10-s + (−1.66 + 0.961i)11-s + (0.402 − 0.697i)13-s + (−1.20 − 0.692i)14-s + (−0.125 − 0.216i)16-s + 0.809i·17-s − 1.13·19-s + (0.531 − 0.307i)20-s + (−0.679 + 1.17i)22-s + (−0.186 − 0.107i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0871i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.996 - 0.0871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.9187716092\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9187716092\) |
| \(L(4)\) |
|
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 | \( 1 + (-132. - 76.7i)T + (7.81e3 + 1.35e4i)T^{2} \) |
| 7 | \( 1 + (336. + 582. i)T + (-5.88e4 + 1.01e5i)T^{2} \) |
| 11 | \( 1 + (2.21e3 - 1.27e3i)T + (8.85e5 - 1.53e6i)T^{2} \) |
| 13 | \( 1 + (-885. + 1.53e3i)T + (-2.41e6 - 4.18e6i)T^{2} \) |
| 17 | \( 1 - 3.97e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 7.76e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + (2.26e3 + 1.30e3i)T + (7.40e7 + 1.28e8i)T^{2} \) |
| 29 | \( 1 + (1.25e4 - 7.23e3i)T + (2.97e8 - 5.15e8i)T^{2} \) |
| 31 | \( 1 + (-2.22e4 + 3.84e4i)T + (-4.43e8 - 7.68e8i)T^{2} \) |
| 37 | \( 1 + 3.95e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + (-1.41e4 - 8.15e3i)T + (2.37e9 + 4.11e9i)T^{2} \) |
| 43 | \( 1 + (2.76e4 + 4.78e4i)T + (-3.16e9 + 5.47e9i)T^{2} \) |
| 47 | \( 1 + (1.03e5 - 5.96e4i)T + (5.38e9 - 9.33e9i)T^{2} \) |
| 53 | \( 1 + 5.93e3iT - 2.21e10T^{2} \) |
| 59 | \( 1 + (6.43e3 + 3.71e3i)T + (2.10e10 + 3.65e10i)T^{2} \) |
| 61 | \( 1 + (2.72e4 + 4.72e4i)T + (-2.57e10 + 4.46e10i)T^{2} \) |
| 67 | \( 1 + (-1.11e5 + 1.93e5i)T + (-4.52e10 - 7.83e10i)T^{2} \) |
| 71 | \( 1 + 7.24e4iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 3.17e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + (8.64e4 + 1.49e5i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 + (-1.26e4 + 7.33e3i)T + (1.63e11 - 2.83e11i)T^{2} \) |
| 89 | \( 1 + 8.00e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + (-8.39e4 - 1.45e5i)T + (-4.16e11 + 7.21e11i)T^{2} \) |
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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
−10.79727783425906348071449620695, −10.34495245385181087391646970052, −9.852572458626643136854679009449, −7.83823212792962089656513636907, −6.74829012136950620008528149284, −5.87307698224924034693898266057, −4.42064504676632944787569637174, −3.17343239105834941622874869657, −1.97459679587121924297873192027, −0.18706353721051567005751132866,
2.14071407719404987935548485310, 3.06232399836096895585774715769, 5.07934437311357153062517586569, 5.72532597997501868834827877000, 6.53819581819374285939756411793, 8.379882472614011165934779294378, 9.025011329551178699590245103500, 10.13277068726139442665218511934, 11.52320814346696211880877122047, 12.62830315790318488571471274841
