| L(s) = 1 | + (282. + 489. i)2-s + (1.10e4 − 1.90e4i)3-s + (−9.41e4 + 1.62e5i)4-s + (−2.74e5 − 4.75e5i)5-s + 1.24e7·6-s + (−6.98e6 − 1.35e7i)7-s − 3.22e7·8-s + (−1.78e8 − 3.08e8i)9-s + (1.55e8 − 2.68e8i)10-s + (6.19e7 − 1.07e8i)11-s + (2.07e9 + 3.59e9i)12-s + 2.37e9·13-s + (4.65e9 − 7.24e9i)14-s − 1.21e10·15-s + (3.21e9 + 5.56e9i)16-s + (−3.28e9 + 5.69e9i)17-s + ⋯ |
| L(s) = 1 | + (0.780 + 1.35i)2-s + (0.969 − 1.67i)3-s + (−0.717 + 1.24i)4-s + (−0.314 − 0.544i)5-s + 3.02·6-s + (−0.458 − 0.888i)7-s − 0.680·8-s + (−1.38 − 2.39i)9-s + (0.490 − 0.849i)10-s + (0.0871 − 0.150i)11-s + (1.39 + 2.41i)12-s + 0.809·13-s + (0.843 − 1.31i)14-s − 1.21·15-s + (0.187 + 0.324i)16-s + (−0.114 + 0.197i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(3.04332 - 0.999438i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.04332 - 0.999438i\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (6.98e6 + 1.35e7i)T \) |
| good | 2 | \( 1 + (-282. - 489. i)T + (-6.55e4 + 1.13e5i)T^{2} \) |
| 3 | \( 1 + (-1.10e4 + 1.90e4i)T + (-6.45e7 - 1.11e8i)T^{2} \) |
| 5 | \( 1 + (2.74e5 + 4.75e5i)T + (-3.81e11 + 6.60e11i)T^{2} \) |
| 11 | \( 1 + (-6.19e7 + 1.07e8i)T + (-2.52e17 - 4.37e17i)T^{2} \) |
| 13 | \( 1 - 2.37e9T + 8.65e18T^{2} \) |
| 17 | \( 1 + (3.28e9 - 5.69e9i)T + (-4.13e20 - 7.16e20i)T^{2} \) |
| 19 | \( 1 + (-4.41e10 - 7.64e10i)T + (-2.74e21 + 4.74e21i)T^{2} \) |
| 23 | \( 1 + (-9.89e10 - 1.71e11i)T + (-7.05e22 + 1.22e23i)T^{2} \) |
| 29 | \( 1 - 3.09e12T + 7.25e24T^{2} \) |
| 31 | \( 1 + (-2.89e10 + 5.01e10i)T + (-1.12e25 - 1.95e25i)T^{2} \) |
| 37 | \( 1 + (1.21e13 + 2.11e13i)T + (-2.28e26 + 3.95e26i)T^{2} \) |
| 41 | \( 1 - 1.39e13T + 2.61e27T^{2} \) |
| 43 | \( 1 + 1.30e13T + 5.87e27T^{2} \) |
| 47 | \( 1 + (-5.50e13 - 9.53e13i)T + (-1.33e28 + 2.30e28i)T^{2} \) |
| 53 | \( 1 + (3.94e13 - 6.83e13i)T + (-1.02e29 - 1.77e29i)T^{2} \) |
| 59 | \( 1 + (-6.64e14 + 1.15e15i)T + (-6.35e29 - 1.10e30i)T^{2} \) |
| 61 | \( 1 + (-6.98e14 - 1.21e15i)T + (-1.12e30 + 1.94e30i)T^{2} \) |
| 67 | \( 1 + (-7.25e14 + 1.25e15i)T + (-5.52e30 - 9.56e30i)T^{2} \) |
| 71 | \( 1 + 2.96e15T + 2.96e31T^{2} \) |
| 73 | \( 1 + (4.39e15 - 7.60e15i)T + (-2.37e31 - 4.11e31i)T^{2} \) |
| 79 | \( 1 + (5.31e15 + 9.20e15i)T + (-9.09e31 + 1.57e32i)T^{2} \) |
| 83 | \( 1 - 3.04e16T + 4.21e32T^{2} \) |
| 89 | \( 1 + (-2.49e16 - 4.31e16i)T + (-6.89e32 + 1.19e33i)T^{2} \) |
| 97 | \( 1 - 4.40e16T + 5.95e33T^{2} \) |
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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
−17.62803131384996543880730888319, −16.10303194463889948269515392444, −14.33561539759777812090438492265, −13.51745518991295990261873038493, −12.48969391038483820415707173089, −8.480179764609073167443322810895, −7.41350820883326934156543943682, −6.23105132376533014666348024375, −3.64527012403125727710274369366, −1.06363254414614963092071479507,
2.63754908041228705263381953852, 3.45328470304844053731518385302, 4.91574701630456641654536480865, 8.939728438221920954273803009539, 10.27243070874433971892655493533, 11.46167710813273980626936734741, 13.54844313005718036520707496043, 14.87981374001465310985585815404, 15.94124419139135026585701047016, 18.99626374009091914839450699494
