| L(s) = 1 | + (−2 − 3.46i)2-s + (15.5 − 1.51i)3-s + (−7.99 + 13.8i)4-s + (33.0 − 57.2i)5-s + (−36.2 − 50.7i)6-s + (−57.0 − 98.8i)7-s + 63.9·8-s + (238. − 47.0i)9-s − 264.·10-s + (192. + 333. i)11-s + (−103. + 227. i)12-s + (−516. + 894. i)13-s + (−228. + 395. i)14-s + (425. − 938. i)15-s + (−128 − 221. i)16-s + 959.·17-s + ⋯ |
| L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.995 − 0.0973i)3-s + (−0.249 + 0.433i)4-s + (0.591 − 1.02i)5-s + (−0.411 − 0.575i)6-s + (−0.440 − 0.762i)7-s + 0.353·8-s + (0.981 − 0.193i)9-s − 0.835·10-s + (0.480 + 0.832i)11-s + (−0.206 + 0.455i)12-s + (−0.847 + 1.46i)13-s + (−0.311 + 0.539i)14-s + (0.488 − 1.07i)15-s + (−0.125 − 0.216i)16-s + 0.804·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.361 + 0.932i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.361 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.27812 - 0.875658i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.27812 - 0.875658i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2 + 3.46i)T \) |
| 3 | \( 1 + (-15.5 + 1.51i)T \) |
| good | 5 | \( 1 + (-33.0 + 57.2i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 7 | \( 1 + (57.0 + 98.8i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 + (-192. - 333. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + (516. - 894. i)T + (-1.85e5 - 3.21e5i)T^{2} \) |
| 17 | \( 1 - 959.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 464.T + 2.47e6T^{2} \) |
| 23 | \( 1 + (1.15e3 - 1.99e3i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + (3.54e3 + 6.14e3i)T + (-1.02e7 + 1.77e7i)T^{2} \) |
| 31 | \( 1 + (3.88e3 - 6.72e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 - 9.31e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (-6.66e3 + 1.15e4i)T + (-5.79e7 - 1.00e8i)T^{2} \) |
| 43 | \( 1 + (-1.05e3 - 1.82e3i)T + (-7.35e7 + 1.27e8i)T^{2} \) |
| 47 | \( 1 + (1.24e3 + 2.16e3i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 + 1.00e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + (2.72e3 - 4.71e3i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (1.70e4 + 2.96e4i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (2.67e4 - 4.64e4i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 + 970.T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.24e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + (1.60e4 + 2.78e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-1.80e4 - 3.12e4i)T + (-1.96e9 + 3.41e9i)T^{2} \) |
| 89 | \( 1 - 4.26e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (-2.19e4 - 3.79e4i)T + (-4.29e9 + 7.43e9i)T^{2} \) |
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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
−17.38545921243229792090258146965, −16.42291179109022691283834270906, −14.41991161420366995179511261574, −13.29475160753135191812451146277, −12.18808942070390514921159856707, −9.836308933444155697069781855300, −9.191918292527484390043958788067, −7.35859196614345558665499002618, −4.20451398805836135541476945585, −1.71573879529527477054922217364,
2.87567830729649205634322786998, 6.00974972257582840385169347036, 7.73228362083177067645477637331, 9.281135919991846334575328180449, 10.44235217733594462925605484253, 12.89582183156047184297847826145, 14.44819586558203577288401118295, 14.97689443725749984377751969127, 16.47287010372745359830104582567, 18.17020529499408258367594837857
