| 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
−18.17020529499408258367594837857, −16.47287010372745359830104582567, −14.97689443725749984377751969127, −14.44819586558203577288401118295, −12.89582183156047184297847826145, −10.44235217733594462925605484253, −9.281135919991846334575328180449, −7.73228362083177067645477637331, −6.00974972257582840385169347036, −2.87567830729649205634322786998,
1.71573879529527477054922217364, 4.20451398805836135541476945585, 7.35859196614345558665499002618, 9.191918292527484390043958788067, 9.836308933444155697069781855300, 12.18808942070390514921159856707, 13.29475160753135191812451146277, 14.41991161420366995179511261574, 16.42291179109022691283834270906, 17.38545921243229792090258146965
