| L(s) = 1 | + (2.73 − 0.731i)2-s + (−21.3 + 79.7i)3-s + (−103. + 60.0i)4-s + (13.2 + 279. i)5-s + 233. i·6-s + (906. − 32.5i)7-s + (−495. + 495. i)8-s + (−4.00e3 − 2.31e3i)9-s + (240. + 752. i)10-s + (−2.46e3 − 4.26e3i)11-s + (−2.56e3 − 9.56e3i)12-s + (9.90e3 + 9.90e3i)13-s + (2.45e3 − 752. i)14-s + (−2.25e4 − 4.90e3i)15-s + (6.69e3 − 1.15e4i)16-s + (−4.47e3 − 1.20e3i)17-s + ⋯ |
| L(s) = 1 | + (0.241 − 0.0646i)2-s + (−0.456 + 1.70i)3-s + (−0.811 + 0.468i)4-s + (0.0475 + 0.998i)5-s + 0.440i·6-s + (0.999 − 0.0358i)7-s + (−0.342 + 0.342i)8-s + (−1.83 − 1.05i)9-s + (0.0760 + 0.237i)10-s + (−0.558 − 0.966i)11-s + (−0.428 − 1.59i)12-s + (1.25 + 1.25i)13-s + (0.238 − 0.0732i)14-s + (−1.72 − 0.375i)15-s + (0.408 − 0.707i)16-s + (−0.221 − 0.0592i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 + 0.488i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.270621 - 1.03769i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.270621 - 1.03769i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-13.2 - 279. i)T \) |
| 7 | \( 1 + (-906. + 32.5i)T \) |
| good | 2 | \( 1 + (-2.73 + 0.731i)T + (110. - 64i)T^{2} \) |
| 3 | \( 1 + (21.3 - 79.7i)T + (-1.89e3 - 1.09e3i)T^{2} \) |
| 11 | \( 1 + (2.46e3 + 4.26e3i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + (-9.90e3 - 9.90e3i)T + 6.27e7iT^{2} \) |
| 17 | \( 1 + (4.47e3 + 1.20e3i)T + (3.55e8 + 2.05e8i)T^{2} \) |
| 19 | \( 1 + (3.69e3 - 6.39e3i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (-1.04e4 - 3.88e4i)T + (-2.94e9 + 1.70e9i)T^{2} \) |
| 29 | \( 1 + 7.26e4iT - 1.72e10T^{2} \) |
| 31 | \( 1 + (6.90e4 - 3.98e4i)T + (1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 + (-1.45e5 + 3.89e4i)T + (8.22e10 - 4.74e10i)T^{2} \) |
| 41 | \( 1 - 4.27e5iT - 1.94e11T^{2} \) |
| 43 | \( 1 + (3.47e4 - 3.47e4i)T - 2.71e11iT^{2} \) |
| 47 | \( 1 + (-7.93e4 - 2.96e5i)T + (-4.38e11 + 2.53e11i)T^{2} \) |
| 53 | \( 1 + (1.72e6 + 4.62e5i)T + (1.01e12 + 5.87e11i)T^{2} \) |
| 59 | \( 1 + (9.72e4 + 1.68e5i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (2.08e6 + 1.20e6i)T + (1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (6.93e5 - 2.58e6i)T + (-5.24e12 - 3.03e12i)T^{2} \) |
| 71 | \( 1 - 3.25e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + (9.50e5 - 3.54e6i)T + (-9.56e12 - 5.52e12i)T^{2} \) |
| 79 | \( 1 + (-3.03e6 - 1.74e6i)T + (9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (3.34e6 + 3.34e6i)T + 2.71e13iT^{2} \) |
| 89 | \( 1 + (3.09e6 - 5.36e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + (3.08e6 - 3.08e6i)T - 8.07e13iT^{2} \) |
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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
−15.67539780147376298296445589397, −14.49369682341326633578991871550, −13.73673981612450602664580360686, −11.40411663727341350211545890345, −11.00804015615089623179492969707, −9.512572560792585354258323544277, −8.318445768340918150718769558611, −5.83785170475260712779677912593, −4.45317977053212170056265304285, −3.39439035417829095261216047792,
0.51496683209439505654628910956, 1.62246557310775799402923209579, 4.88703971691184602427377792000, 5.92237275365203009074636787785, 7.77160782873341030035447319234, 8.705651973314509635593946175050, 10.80270573221972930283248928017, 12.39192972954306165899859730820, 12.99622080734955578645836410935, 13.85424978980575552855772850656
