| L(s) = 1 | + (0.837 − 0.147i)2-s + (0.252 − 2.98i)3-s + (−3.07 + 1.12i)4-s + (−3.55 − 4.23i)5-s + (−0.229 − 2.54i)6-s + (0.434 + 0.751i)7-s + (−5.35 + 3.09i)8-s + (−8.87 − 1.51i)9-s + (−3.60 − 3.02i)10-s + 19.9i·11-s + (2.57 + 9.48i)12-s + (−7.38 − 6.19i)13-s + (0.474 + 0.565i)14-s + (−13.5 + 9.55i)15-s + (6.00 − 5.04i)16-s + (−13.9 − 16.6i)17-s + ⋯ |
| L(s) = 1 | + (0.418 − 0.0738i)2-s + (0.0843 − 0.996i)3-s + (−0.769 + 0.280i)4-s + (−0.710 − 0.847i)5-s + (−0.0382 − 0.423i)6-s + (0.0620 + 0.107i)7-s + (−0.669 + 0.386i)8-s + (−0.985 − 0.168i)9-s + (−0.360 − 0.302i)10-s + 1.81i·11-s + (0.214 + 0.790i)12-s + (−0.567 − 0.476i)13-s + (0.0339 + 0.0404i)14-s + (−0.904 + 0.637i)15-s + (0.375 − 0.315i)16-s + (−0.823 − 0.980i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.104i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.994 - 0.104i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0273612 + 0.521469i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0273612 + 0.521469i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.252 + 2.98i)T \) |
| 19 | \( 1 + (1.50 + 18.9i)T \) |
| good | 2 | \( 1 + (-0.837 + 0.147i)T + (3.75 - 1.36i)T^{2} \) |
| 5 | \( 1 + (3.55 + 4.23i)T + (-4.34 + 24.6i)T^{2} \) |
| 7 | \( 1 + (-0.434 - 0.751i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 - 19.9iT - 121T^{2} \) |
| 13 | \( 1 + (7.38 + 6.19i)T + (29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (13.9 + 16.6i)T + (-50.1 + 284. i)T^{2} \) |
| 23 | \( 1 + (1.54 + 4.25i)T + (-405. + 340. i)T^{2} \) |
| 29 | \( 1 + (12.0 + 33.0i)T + (-644. + 540. i)T^{2} \) |
| 31 | \( 1 - 20.6T + 961T^{2} \) |
| 37 | \( 1 - 9.72T + 1.36e3T^{2} \) |
| 41 | \( 1 + (67.5 - 11.9i)T + (1.57e3 - 574. i)T^{2} \) |
| 43 | \( 1 + (12.8 + 4.69i)T + (1.41e3 + 1.18e3i)T^{2} \) |
| 47 | \( 1 + (-10.5 - 28.9i)T + (-1.69e3 + 1.41e3i)T^{2} \) |
| 53 | \( 1 + (8.23 + 1.45i)T + (2.63e3 + 960. i)T^{2} \) |
| 59 | \( 1 + (3.25 - 8.93i)T + (-2.66e3 - 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-85.2 - 71.5i)T + (646. + 3.66e3i)T^{2} \) |
| 67 | \( 1 + (-15.6 + 88.9i)T + (-4.21e3 - 1.53e3i)T^{2} \) |
| 71 | \( 1 + (75.8 - 13.3i)T + (4.73e3 - 1.72e3i)T^{2} \) |
| 73 | \( 1 + (-24.1 - 8.77i)T + (4.08e3 + 3.42e3i)T^{2} \) |
| 79 | \( 1 + (79.4 - 66.6i)T + (1.08e3 - 6.14e3i)T^{2} \) |
| 83 | \( 1 + (-117. + 67.7i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (-8.35 - 22.9i)T + (-6.06e3 + 5.09e3i)T^{2} \) |
| 97 | \( 1 + (-4.90 - 27.8i)T + (-8.84e3 + 3.21e3i)T^{2} \) |
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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
−12.14168582642772884048693893912, −11.70879741184838068493538760600, −9.739769527155204386803242840991, −8.771664297041680185659966172939, −7.86101395696124896957152853423, −6.91192779878985312015496343537, −5.12030426442394224683190144691, −4.39180833232733191382452918648, −2.51040913862384054079117765021, −0.27218378333402887082952272652,
3.31824399026934728504899353865, 3.99501177862827739678959373173, 5.34940284116653137490266459268, 6.43523203116311979352875855068, 8.201095745811150457333529408167, 8.940020778625535551330152350968, 10.21495891882453155782486673417, 10.94259110444583618286749055401, 11.85822229182971101021651892387, 13.32078630800277868259936090525
