| L(s) = 1 | + i·3-s − 4.73i·7-s − 9-s + 4.34·11-s + 2.30i·13-s − 0.618i·17-s − 4.41·19-s + 4.73·21-s + 6.34i·23-s − i·27-s − 7.54·29-s − 4.80·31-s + 4.34i·33-s + 10.9i·37-s − 2.30·39-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 1.78i·7-s − 0.333·9-s + 1.31·11-s + 0.639i·13-s − 0.149i·17-s − 1.01·19-s + 1.03·21-s + 1.32i·23-s − 0.192i·27-s − 1.40·29-s − 0.862·31-s + 0.757i·33-s + 1.80i·37-s − 0.369·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.374191601\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.374191601\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 4.73iT - 7T^{2} \) |
| 11 | \( 1 - 4.34T + 11T^{2} \) |
| 13 | \( 1 - 2.30iT - 13T^{2} \) |
| 17 | \( 1 + 0.618iT - 17T^{2} \) |
| 19 | \( 1 + 4.41T + 19T^{2} \) |
| 23 | \( 1 - 6.34iT - 23T^{2} \) |
| 29 | \( 1 + 7.54T + 29T^{2} \) |
| 31 | \( 1 + 4.80T + 31T^{2} \) |
| 37 | \( 1 - 10.9iT - 37T^{2} \) |
| 41 | \( 1 + 5.20T + 41T^{2} \) |
| 43 | \( 1 - 6.85iT - 43T^{2} \) |
| 47 | \( 1 - 0.165iT - 47T^{2} \) |
| 53 | \( 1 + 2.16iT - 53T^{2} \) |
| 59 | \( 1 - 12.9T + 59T^{2} \) |
| 61 | \( 1 - 4.11T + 61T^{2} \) |
| 67 | \( 1 - 14.6iT - 67T^{2} \) |
| 71 | \( 1 - 4.92T + 71T^{2} \) |
| 73 | \( 1 + 7.91iT - 73T^{2} \) |
| 79 | \( 1 - 17.0T + 79T^{2} \) |
| 83 | \( 1 + 5.72iT - 83T^{2} \) |
| 89 | \( 1 + 0.816T + 89T^{2} \) |
| 97 | \( 1 - 7.20iT - 97T^{2} \) |
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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
−8.249287434840042760925624310520, −7.45726976621930356317286583659, −6.82242372797424890503421928186, −6.33641124724757154733829165490, −5.23171415651609678919225846582, −4.44331320456036136736382275229, −3.79477051494600534808891857336, −3.49570423609070497626109344516, −1.90832263229045716149693332363, −1.05984166767295396924635449964,
0.37256297523656360201148427452, 1.92954107625152772597088603176, 2.23202021655212948037062778437, 3.37653686068264132714555389966, 4.16139647712172100846634689232, 5.36999605552138083515376080420, 5.71628195765955340129685397085, 6.50540050809015695334060862176, 7.02976732694934975626017225436, 8.089358623624594872353466465524
