| L(s) = 1 | + (−1 + 2i)5-s + 2i·7-s + 5·11-s − 4i·13-s − 6i·17-s + 7·19-s − 4i·23-s + (−3 − 4i)25-s − 5·29-s − 3·31-s + (−4 − 2i)35-s − 2i·37-s − 7·41-s − 6i·43-s − 6i·47-s + ⋯ |
| L(s) = 1 | + (−0.447 + 0.894i)5-s + 0.755i·7-s + 1.50·11-s − 1.10i·13-s − 1.45i·17-s + 1.60·19-s − 0.834i·23-s + (−0.600 − 0.800i)25-s − 0.928·29-s − 0.538·31-s + (−0.676 − 0.338i)35-s − 0.328i·37-s − 1.09·41-s − 0.914i·43-s − 0.875i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.731452527\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.731452527\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1 - 2i)T \) |
| good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 - 5T + 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 - 7T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 7T + 41T^{2} \) |
| 43 | \( 1 + 6iT - 43T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 - 10iT - 53T^{2} \) |
| 59 | \( 1 + 15T + 59T^{2} \) |
| 61 | \( 1 - 14T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 - 5T + 71T^{2} \) |
| 73 | \( 1 + 14iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 14iT - 83T^{2} \) |
| 89 | \( 1 - 3T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−8.739766977327316168551549030044, −7.59925169840168080336403985486, −7.25816312128100823202082167354, −6.37484696475546100397345377836, −5.61749562875058607116179341784, −4.82508550326625055024023385626, −3.59387681436439858374163140986, −3.14899033437529600440979433568, −2.10778927255725965753624548073, −0.61838830697776382225402053268,
1.15113616788828396308145435499, 1.68918136824575440729859233700, 3.72212784264987148652800031930, 3.74976450356557121483933309561, 4.75515824880593334908041869324, 5.64194709881183865967712858245, 6.56678118655913522222147313566, 7.23408295871254035710001612488, 7.973103881985362746939187645620, 8.751247200358200256677502074403
