| L(s) = 1 | + 3.14i·3-s + (−2.07 + 0.844i)5-s + 1.20i·7-s − 6.86·9-s − 3.65·11-s − 0.859i·13-s + (−2.65 − 6.50i)15-s + 6.72i·17-s − 1.51·19-s − 3.78·21-s − i·23-s + (3.57 − 3.49i)25-s − 12.1i·27-s − 0.548·29-s + 5.99·31-s + ⋯ |
| L(s) = 1 | + 1.81i·3-s + (−0.925 + 0.377i)5-s + 0.456i·7-s − 2.28·9-s − 1.10·11-s − 0.238i·13-s + (−0.685 − 1.67i)15-s + 1.63i·17-s − 0.348·19-s − 0.826·21-s − 0.208i·23-s + (0.714 − 0.699i)25-s − 2.33i·27-s − 0.101·29-s + 1.07·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.377 + 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.377 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2561724765\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2561724765\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (2.07 - 0.844i)T \) |
| 23 | \( 1 + iT \) |
| good | 3 | \( 1 - 3.14iT - 3T^{2} \) |
| 7 | \( 1 - 1.20iT - 7T^{2} \) |
| 11 | \( 1 + 3.65T + 11T^{2} \) |
| 13 | \( 1 + 0.859iT - 13T^{2} \) |
| 17 | \( 1 - 6.72iT - 17T^{2} \) |
| 19 | \( 1 + 1.51T + 19T^{2} \) |
| 29 | \( 1 + 0.548T + 29T^{2} \) |
| 31 | \( 1 - 5.99T + 31T^{2} \) |
| 37 | \( 1 - 2.04iT - 37T^{2} \) |
| 41 | \( 1 + 7.14T + 41T^{2} \) |
| 43 | \( 1 + 10.0iT - 43T^{2} \) |
| 47 | \( 1 - 9.17iT - 47T^{2} \) |
| 53 | \( 1 + 5.37iT - 53T^{2} \) |
| 59 | \( 1 + 0.582T + 59T^{2} \) |
| 61 | \( 1 + 8.83T + 61T^{2} \) |
| 67 | \( 1 - 3.20iT - 67T^{2} \) |
| 71 | \( 1 - 12.5T + 71T^{2} \) |
| 73 | \( 1 + 8.62iT - 73T^{2} \) |
| 79 | \( 1 - 0.0700T + 79T^{2} \) |
| 83 | \( 1 - 6.74iT - 83T^{2} \) |
| 89 | \( 1 + 4.96T + 89T^{2} \) |
| 97 | \( 1 - 11.3iT - 97T^{2} \) |
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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
−10.15494203314246434614183590401, −9.097050008003390296712818737602, −8.334458725442572272805972839680, −7.963909722715058736137369924350, −6.54953822712274605328706840191, −5.63750725135761927495166431320, −4.87090340295424459871274726466, −4.08789794647319528268330710447, −3.38047141408683041205908817689, −2.51134881634361145861186343381,
0.10807177673584569866187233578, 1.05653088922196797732773567156, 2.38139494358282878782700996908, 3.21644645913716264930102700117, 4.61632861642795813487565477385, 5.42116429835884118129460976360, 6.52733016548332903894545437750, 7.21246051020905375314842259488, 7.70062612557075526045967768770, 8.287048738131426165188189341238
