| L(s) = 1 | + (0.326 + 1.37i)2-s + (2.88 − 0.816i)3-s + (−1.78 + 0.897i)4-s + (6.99 − 5.20i)5-s + (2.06 + 3.70i)6-s + (−4.06 + 2.67i)7-s + (−1.81 − 2.16i)8-s + (7.66 − 4.71i)9-s + (9.44 + 7.92i)10-s + (−0.175 + 1.49i)11-s + (−4.42 + 4.05i)12-s + (−4.03 − 13.4i)13-s + (−5.00 − 4.72i)14-s + (15.9 − 20.7i)15-s + (2.38 − 3.20i)16-s + (3.91 + 0.690i)17-s + ⋯ |
| L(s) = 1 | + (0.163 + 0.688i)2-s + (0.962 − 0.272i)3-s + (−0.446 + 0.224i)4-s + (1.39 − 1.04i)5-s + (0.344 + 0.617i)6-s + (−0.581 + 0.382i)7-s + (−0.227 − 0.270i)8-s + (0.851 − 0.523i)9-s + (0.944 + 0.792i)10-s + (−0.0159 + 0.136i)11-s + (−0.368 + 0.337i)12-s + (−0.310 − 1.03i)13-s + (−0.357 − 0.337i)14-s + (1.06 − 1.38i)15-s + (0.149 − 0.200i)16-s + (0.230 + 0.0406i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 - 0.246i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.969 - 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.31805 + 0.290682i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.31805 + 0.290682i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.326 - 1.37i)T \) |
| 3 | \( 1 + (-2.88 + 0.816i)T \) |
| good | 5 | \( 1 + (-6.99 + 5.20i)T + (7.17 - 23.9i)T^{2} \) |
| 7 | \( 1 + (4.06 - 2.67i)T + (19.4 - 44.9i)T^{2} \) |
| 11 | \( 1 + (0.175 - 1.49i)T + (-117. - 27.9i)T^{2} \) |
| 13 | \( 1 + (4.03 + 13.4i)T + (-141. + 92.8i)T^{2} \) |
| 17 | \( 1 + (-3.91 - 0.690i)T + (271. + 98.8i)T^{2} \) |
| 19 | \( 1 + (-5.84 - 33.1i)T + (-339. + 123. i)T^{2} \) |
| 23 | \( 1 + (21.1 - 32.1i)T + (-209. - 485. i)T^{2} \) |
| 29 | \( 1 + (31.6 - 29.8i)T + (48.8 - 839. i)T^{2} \) |
| 31 | \( 1 + (-0.170 + 2.92i)T + (-954. - 111. i)T^{2} \) |
| 37 | \( 1 + (-12.9 + 4.69i)T + (1.04e3 - 879. i)T^{2} \) |
| 41 | \( 1 + (3.69 - 15.6i)T + (-1.50e3 - 754. i)T^{2} \) |
| 43 | \( 1 + (25.5 + 59.1i)T + (-1.26e3 + 1.34e3i)T^{2} \) |
| 47 | \( 1 + (-29.9 + 1.74i)T + (2.19e3 - 256. i)T^{2} \) |
| 53 | \( 1 + (54.7 + 31.6i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (7.55 + 64.6i)T + (-3.38e3 + 802. i)T^{2} \) |
| 61 | \( 1 + (15.5 + 7.79i)T + (2.22e3 + 2.98e3i)T^{2} \) |
| 67 | \( 1 + (-68.2 + 72.3i)T + (-261. - 4.48e3i)T^{2} \) |
| 71 | \( 1 + (18.0 - 21.5i)T + (-875. - 4.96e3i)T^{2} \) |
| 73 | \( 1 + (49.3 - 41.4i)T + (925. - 5.24e3i)T^{2} \) |
| 79 | \( 1 + (98.2 - 23.2i)T + (5.57e3 - 2.80e3i)T^{2} \) |
| 83 | \( 1 + (-17.4 - 73.4i)T + (-6.15e3 + 3.09e3i)T^{2} \) |
| 89 | \( 1 + (-37.2 - 44.3i)T + (-1.37e3 + 7.80e3i)T^{2} \) |
| 97 | \( 1 + (6.91 - 9.28i)T + (-2.69e3 - 9.01e3i)T^{2} \) |
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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
−12.77692864098125010242689850822, −12.44040294494472660864693371897, −9.930515715204335307804896174767, −9.596681910148090655946661312375, −8.513736641719494664191535390009, −7.59051974445968714110535425108, −6.05390115716887599319872920670, −5.32394943266967708661208266372, −3.50124198733348810779020142110, −1.73419456051361295380915026403,
2.13028750335764869492075026797, 3.00515414027141616974532507942, 4.48005331293412296094749461124, 6.21047800575135507812496694696, 7.23669677714445051359214701674, 8.980791524862147943278061874433, 9.718684294758339411275264143238, 10.32416137158769736247540833755, 11.37830881742310275136932794329, 12.98767879535669382365630616218
