| L(s) = 1 | + (−0.248 + 1.98i)2-s + (−1.21 − 2.74i)3-s + (−3.87 − 0.987i)4-s + (−0.719 − 4.07i)5-s + (5.74 − 1.71i)6-s + (0.106 + 0.0891i)7-s + (2.92 − 7.44i)8-s + (−6.07 + 6.64i)9-s + (8.27 − 0.412i)10-s + (−3.25 + 18.4i)11-s + (1.98 + 11.8i)12-s + (−5.32 + 14.6i)13-s + (−0.203 + 0.188i)14-s + (−10.3 + 6.91i)15-s + (14.0 + 7.65i)16-s + (−1.59 + 0.922i)17-s + ⋯ |
| L(s) = 1 | + (−0.124 + 0.992i)2-s + (−0.403 − 0.915i)3-s + (−0.969 − 0.246i)4-s + (−0.143 − 0.815i)5-s + (0.958 − 0.286i)6-s + (0.0151 + 0.0127i)7-s + (0.365 − 0.930i)8-s + (−0.674 + 0.738i)9-s + (0.827 − 0.0412i)10-s + (−0.296 + 1.68i)11-s + (0.165 + 0.986i)12-s + (−0.409 + 1.12i)13-s + (−0.0145 + 0.0134i)14-s + (−0.688 + 0.460i)15-s + (0.878 + 0.478i)16-s + (−0.0939 + 0.0542i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.917 - 0.397i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.917 - 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0719798 + 0.347615i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0719798 + 0.347615i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.248 - 1.98i)T \) |
| 3 | \( 1 + (1.21 + 2.74i)T \) |
| good | 5 | \( 1 + (0.719 + 4.07i)T + (-23.4 + 8.55i)T^{2} \) |
| 7 | \( 1 + (-0.106 - 0.0891i)T + (8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (3.25 - 18.4i)T + (-113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (5.32 - 14.6i)T + (-129. - 108. i)T^{2} \) |
| 17 | \( 1 + (1.59 - 0.922i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (25.5 + 14.7i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-12.8 - 15.3i)T + (-91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (-14.9 + 5.43i)T + (644. - 540. i)T^{2} \) |
| 31 | \( 1 + (44.5 - 37.4i)T + (166. - 946. i)T^{2} \) |
| 37 | \( 1 + (15.6 - 9.02i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-9.34 + 25.6i)T + (-1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (70.2 + 12.3i)T + (1.73e3 + 632. i)T^{2} \) |
| 47 | \( 1 + (31.7 - 37.8i)T + (-383. - 2.17e3i)T^{2} \) |
| 53 | \( 1 - 52.5T + 2.80e3T^{2} \) |
| 59 | \( 1 + (4.51 + 25.6i)T + (-3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (34.3 - 40.8i)T + (-646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (27.2 - 74.8i)T + (-3.43e3 - 2.88e3i)T^{2} \) |
| 71 | \( 1 + (6.09 - 3.51i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (5.25 - 9.10i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-68.1 + 24.7i)T + (4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (24.4 - 8.90i)T + (5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (7.40 + 4.27i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-13.1 + 74.5i)T + (-8.84e3 - 3.21e3i)T^{2} \) |
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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
−12.72442471572613850967079838606, −11.90642231422227722803772790578, −10.45554420505130677525039161084, −9.189675529930860530533310519514, −8.435842374343241929924857313778, −7.15431842929105013774186067322, −6.77606172999424084449653159579, −5.18915832825015181342422939228, −4.53339723711766761888539252440, −1.75677739568715570149519967714,
0.21348154840829899250011895720, 2.86763385770686783286518115704, 3.67911237201872291126249058594, 5.11049529139570866682615694751, 6.19630417804651230994633327970, 8.049666727227042387228309112106, 8.900088891951575025014757982981, 10.16641855135625895959933183252, 10.77936184251133280028354976971, 11.22435187898847135248568325946
