L(s) = 1 | + i·2-s + (−1.61 − 0.618i)3-s − 4-s − 1.23·5-s + (0.618 − 1.61i)6-s + (0.381 + 2.61i)7-s − i·8-s + (2.23 + 2.00i)9-s − 1.23i·10-s + (1.61 + 0.618i)12-s − i·13-s + (−2.61 + 0.381i)14-s + (2.00 + 0.763i)15-s + 16-s − 6.47·17-s + (−2.00 + 2.23i)18-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (−0.934 − 0.356i)3-s − 0.5·4-s − 0.552·5-s + (0.252 − 0.660i)6-s + (0.144 + 0.989i)7-s − 0.353i·8-s + (0.745 + 0.666i)9-s − 0.390i·10-s + (0.467 + 0.178i)12-s − 0.277i·13-s + (−0.699 + 0.102i)14-s + (0.516 + 0.197i)15-s + 0.250·16-s − 1.56·17-s + (−0.471 + 0.527i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.218 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.218 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.137154 - 0.171210i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.137154 - 0.171210i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + (1.61 + 0.618i)T \) |
| 7 | \( 1 + (-0.381 - 2.61i)T \) |
| 13 | \( 1 + iT \) |
good | 5 | \( 1 + 1.23T + 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 17 | \( 1 + 6.47T + 17T^{2} \) |
| 19 | \( 1 + 6iT - 19T^{2} \) |
| 23 | \( 1 + 1.23iT - 23T^{2} \) |
| 29 | \( 1 + 7.70iT - 29T^{2} \) |
| 31 | \( 1 + 2.76iT - 31T^{2} \) |
| 37 | \( 1 + 4.76T + 37T^{2} \) |
| 41 | \( 1 + 0.763T + 41T^{2} \) |
| 43 | \( 1 + 1.52T + 43T^{2} \) |
| 47 | \( 1 - 2.47T + 47T^{2} \) |
| 53 | \( 1 + 1.23iT - 53T^{2} \) |
| 59 | \( 1 - 4.47T + 59T^{2} \) |
| 61 | \( 1 - 4.47iT - 61T^{2} \) |
| 67 | \( 1 + 10.9T + 67T^{2} \) |
| 71 | \( 1 + 5.52iT - 71T^{2} \) |
| 73 | \( 1 + 1.23iT - 73T^{2} \) |
| 79 | \( 1 - 8.94T + 79T^{2} \) |
| 83 | \( 1 - 14T + 83T^{2} \) |
| 89 | \( 1 + 16.1T + 89T^{2} \) |
| 97 | \( 1 - 5.23iT - 97T^{2} \) |
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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
−10.75042178671790385824993338845, −9.511357783977438458691403276512, −8.616643449162671273166745190444, −7.73992541799218099319462437319, −6.78850499279001052125873378114, −6.04893702698268726148281776814, −5.07030838922238432529418584516, −4.23861201714527554878012662128, −2.32638493614287781146696039331, −0.14492481619184129478389763290,
1.54522890431399930719550629383, 3.60470593542013135407300783335, 4.24853661178930998788238765393, 5.21309367697032257563326574976, 6.50943660639169949410297023996, 7.38004250907892275496881373515, 8.542261255176508036876940730888, 9.595382528803111981652385344534, 10.50969770103306145983237414173, 10.94293650779809128354135629313