L(s) = 1 | + i·2-s − i·3-s − 4-s + 2.62·5-s + 6-s − 2.77·7-s − i·8-s − 9-s + 2.62i·10-s + 3.21i·11-s + i·12-s + 6.49·13-s − 2.77i·14-s − 2.62i·15-s + 16-s − 0.0445i·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 1.17·5-s + 0.408·6-s − 1.04·7-s − 0.353i·8-s − 0.333·9-s + 0.829i·10-s + 0.969i·11-s + 0.288i·12-s + 1.80·13-s − 0.742i·14-s − 0.676i·15-s + 0.250·16-s − 0.0108i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3306 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0783 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3306 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0783 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.820434121\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.820434121\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + iT \) |
| 19 | \( 1 - iT \) |
| 29 | \( 1 + (-5.36 - 0.422i)T \) |
good | 5 | \( 1 - 2.62T + 5T^{2} \) |
| 7 | \( 1 + 2.77T + 7T^{2} \) |
| 11 | \( 1 - 3.21iT - 11T^{2} \) |
| 13 | \( 1 - 6.49T + 13T^{2} \) |
| 17 | \( 1 + 0.0445iT - 17T^{2} \) |
| 23 | \( 1 + 6.57T + 23T^{2} \) |
| 31 | \( 1 - 9.01iT - 31T^{2} \) |
| 37 | \( 1 + 4.68iT - 37T^{2} \) |
| 41 | \( 1 - 3.64iT - 41T^{2} \) |
| 43 | \( 1 + 0.727iT - 43T^{2} \) |
| 47 | \( 1 - 0.287iT - 47T^{2} \) |
| 53 | \( 1 - 2.62T + 53T^{2} \) |
| 59 | \( 1 - 13.6T + 59T^{2} \) |
| 61 | \( 1 - 5.87iT - 61T^{2} \) |
| 67 | \( 1 + 8.85T + 67T^{2} \) |
| 71 | \( 1 - 1.84T + 71T^{2} \) |
| 73 | \( 1 + 1.64iT - 73T^{2} \) |
| 79 | \( 1 - 3.13iT - 79T^{2} \) |
| 83 | \( 1 + 4.50T + 83T^{2} \) |
| 89 | \( 1 - 12.4iT - 89T^{2} \) |
| 97 | \( 1 - 10.6iT - 97T^{2} \) |
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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
−8.728234805338613992090822883911, −8.099360134272828835291189297056, −7.03984504033735445055762728987, −6.50938658782042073165040669180, −6.01412173182134355446475356564, −5.39217644442058624819198357883, −4.19386694441901322492805181051, −3.29156580081942462942471941794, −2.13872866338367912841702501116, −1.16697854839377143499240677621,
0.60416748619039539012198844521, 1.88489162484357873818676055328, 2.91530181647514703611397088997, 3.60026251166209569544703235598, 4.33782496367138281751075133679, 5.70034631610099448273103228712, 5.93992918484660142192326027518, 6.59317604585588097714512756079, 8.105415299689415175593485575093, 8.701413565101950769701805512958