L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)4-s − 7-s − 8-s + (−0.5 − 0.866i)11-s + (0.5 + 0.866i)13-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s − 22-s + (0.5 − 0.866i)23-s + 26-s + (0.5 + 0.866i)28-s + (−0.5 − 0.866i)29-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)4-s − 7-s − 8-s + (−0.5 − 0.866i)11-s + (0.5 + 0.866i)13-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s − 22-s + (0.5 − 0.866i)23-s + 26-s + (0.5 + 0.866i)28-s + (−0.5 − 0.866i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1665 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.851 - 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1665 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.851 - 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.286549600 - 0.3637133036i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.286549600 - 0.3637133036i\) |
\(L(1)\) |
\(\approx\) |
\(0.9797787953 - 0.4409392203i\) |
\(L(1)\) |
\(\approx\) |
\(0.9797787953 - 0.4409392203i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 \) |
good | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 + (0.5 + 0.866i)T \) |
| 17 | \( 1 + (0.5 + 0.866i)T \) |
| 19 | \( 1 + (-0.5 + 0.866i)T \) |
| 23 | \( 1 + (0.5 - 0.866i)T \) |
| 29 | \( 1 + (-0.5 - 0.866i)T \) |
| 31 | \( 1 + (-0.5 + 0.866i)T \) |
| 41 | \( 1 + (-0.5 - 0.866i)T \) |
| 43 | \( 1 + (0.5 + 0.866i)T \) |
| 47 | \( 1 + (0.5 + 0.866i)T \) |
| 53 | \( 1 + (0.5 + 0.866i)T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 + (0.5 + 0.866i)T \) |
| 71 | \( 1 + (-0.5 + 0.866i)T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 + (0.5 - 0.866i)T \) |
| 89 | \( 1 + (-0.5 - 0.866i)T \) |
| 97 | \( 1 + (0.5 + 0.866i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−20.53879166893979193101736951230, −19.81749420415329768190077727947, −18.725889249790206212197122962005, −18.116296391900686346760998746255, −17.365819775160170611846692472495, −16.576427906355561632352454928272, −15.8761228954923363840705834725, −15.27556558828472035115282221968, −14.71321908736066070529856344405, −13.43134859164327873650710415583, −13.21698876682846169645769145733, −12.47797165373275826763555354784, −11.59929400201292872645118457648, −10.50166356533954262995933241723, −9.58396720007506114603782465401, −8.991133725634854465554465739487, −7.93939608467609362936498146964, −7.18853326177152536883585473971, −6.651151349666941861609473186324, −5.51855913149507343490986929292, −5.1333892606065074168132873159, −3.91664166640870818238561949919, −3.22400639091542928441067709748, −2.353500228774286127364171113482, −0.52733416805431184115790325309,
0.87182979165095141235418030109, 1.98259673768385658904590940690, 2.9312380210930196007650415550, 3.70546512535366673374954221993, 4.32891615213646590178986819346, 5.68287432515049145275408559348, 6.02369650524037862253746993725, 6.98463601570855912139636871490, 8.37398442689099000064209031924, 8.940186183623059690004779066920, 9.910333851863562780831256299549, 10.53971988836328339305239674111, 11.19754717125620910026204371878, 12.17397960333473293546300082030, 12.78844930417065421089947830388, 13.39855042715568157173489775940, 14.20027679771587411953904710836, 14.86200909192910238453269088226, 15.9106777660915392621313298279, 16.45432061115272498994595940821, 17.385505157531904029785963044976, 18.626858387186279359400442841809, 18.9500312700090269351621014168, 19.38682017375882774341320402389, 20.51538136846960901553428285288