| L(s) = 1 | − i·3-s + 1.79i·5-s − 0.158·7-s − 9-s − 5.37i·11-s + 5.95i·13-s + 1.79·15-s − 3.05·17-s − 3.05i·19-s + 0.158i·21-s − 2.82·23-s + 1.76·25-s + i·27-s + 2.96i·29-s − 4.15·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.804i·5-s − 0.0600·7-s − 0.333·9-s − 1.61i·11-s + 1.65i·13-s + 0.464·15-s − 0.740·17-s − 0.700i·19-s + 0.0346i·21-s − 0.589·23-s + 0.353·25-s + 0.192i·27-s + 0.551i·29-s − 0.746·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1669067760\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1669067760\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| good | 5 | \( 1 - 1.79iT - 5T^{2} \) |
| 7 | \( 1 + 0.158T + 7T^{2} \) |
| 11 | \( 1 + 5.37iT - 11T^{2} \) |
| 13 | \( 1 - 5.95iT - 13T^{2} \) |
| 17 | \( 1 + 3.05T + 17T^{2} \) |
| 19 | \( 1 + 3.05iT - 19T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 - 2.96iT - 29T^{2} \) |
| 31 | \( 1 + 4.15T + 31T^{2} \) |
| 37 | \( 1 + 8.46iT - 37T^{2} \) |
| 41 | \( 1 + 2.60T + 41T^{2} \) |
| 43 | \( 1 - 8.13iT - 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 + 5.03iT - 53T^{2} \) |
| 59 | \( 1 + 5.65iT - 59T^{2} \) |
| 61 | \( 1 - 5.18iT - 61T^{2} \) |
| 67 | \( 1 + 1.08iT - 67T^{2} \) |
| 71 | \( 1 - 0.317T + 71T^{2} \) |
| 73 | \( 1 + 1.33T + 73T^{2} \) |
| 79 | \( 1 + 9.69T + 79T^{2} \) |
| 83 | \( 1 - 0.163iT - 83T^{2} \) |
| 89 | \( 1 + 14.3T + 89T^{2} \) |
| 97 | \( 1 + 0.571T + 97T^{2} \) |
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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
−8.436083379570160432056229661774, −7.42430774809290333522115810534, −6.70428225487454361771356625402, −6.35674675152249793760892308181, −5.42742522550018657008946399778, −4.32112236221023851174646950434, −3.40099366876499198303596006714, −2.58540286079200609609015262359, −1.58033302749923087970096427444, −0.04976240190579107571895710566,
1.48860537023603168447250949542, 2.60493676687538436863132105157, 3.69083490816099777163746688164, 4.54424238319672636949722739493, 5.10481375738572430506687475817, 5.85916014609869821383236220255, 6.86503559871263704018274877697, 7.75557248080051730808196926505, 8.352677612798714540380330624863, 9.089113086852909004289296190116
