| L(s) = 1 | + (−4 + 6.92i)2-s + (−38.4 − 66.5i)3-s + (−31.9 − 55.4i)4-s + (−10.6 + 18.3i)5-s + 615.·6-s + 511.·8-s + (−1.86e3 + 3.22e3i)9-s + (−84.9 − 147. i)10-s + (−1.63e3 − 2.82e3i)11-s + (−2.46e3 + 4.26e3i)12-s − 1.02e4·13-s + 1.63e3·15-s + (−2.04e3 + 3.54e3i)16-s + (−1.82e4 − 3.16e4i)17-s + (−1.48e4 − 2.57e4i)18-s + (−1.62e4 + 2.80e4i)19-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.821 − 1.42i)3-s + (−0.249 − 0.433i)4-s + (−0.0380 + 0.0658i)5-s + 1.16·6-s + 0.353·8-s + (−0.851 + 1.47i)9-s + (−0.0268 − 0.0465i)10-s + (−0.369 − 0.640i)11-s + (−0.410 + 0.711i)12-s − 1.29·13-s + 0.124·15-s + (−0.125 + 0.216i)16-s + (−0.901 − 1.56i)17-s + (−0.601 − 1.04i)18-s + (−0.541 + 0.938i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.366772 + 0.181818i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.366772 + 0.181818i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (4 - 6.92i)T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + (38.4 + 66.5i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 5 | \( 1 + (10.6 - 18.3i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 11 | \( 1 + (1.63e3 + 2.82e3i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + 1.02e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + (1.82e4 + 3.16e4i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 19 | \( 1 + (1.62e4 - 2.80e4i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (8.37e3 - 1.44e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 - 4.39e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + (-6.69e4 - 1.15e5i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 + (-2.89e5 + 5.02e5i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 - 5.32e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 3.65e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (1.03e5 - 1.79e5i)T + (-2.53e11 - 4.38e11i)T^{2} \) |
| 53 | \( 1 + (-5.44e5 - 9.42e5i)T + (-5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 + (-8.08e4 - 1.39e5i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (4.22e5 - 7.31e5i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (6.13e5 + 1.06e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 - 1.10e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + (-8.19e5 - 1.41e6i)T + (-5.52e12 + 9.56e12i)T^{2} \) |
| 79 | \( 1 + (-2.49e6 + 4.32e6i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 - 3.38e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (-3.26e6 + 5.66e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + 1.35e6T + 8.07e13T^{2} \) |
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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
−12.68852115488791401518079924344, −11.72017153527031197121524437603, −10.70262713060261345998409538007, −9.199493583754507960308680650498, −7.77428298743245326718293373991, −7.12224331827465983967772335925, −6.05774028882963220555756374314, −4.98184093002915125733009529958, −2.37888473763935008848970503264, −0.790445255718541798273727434668,
0.22886469838026289738577272730, 2.45956792856504556374339965390, 4.23220562804025440794320760518, 4.85890376309801448767384516786, 6.49525124326847031789132459558, 8.290867386839882595350158230406, 9.555814907869242831915955330450, 10.26286647487319450526735528243, 11.03339693428381217342408919679, 12.07475938133996572829076747393
