| L(s) = 1 | + 3-s + (2 + i)5-s + 4i·7-s + 9-s + 4i·11-s + (2 + i)15-s + 4i·17-s + 4i·21-s − 4i·23-s + (3 + 4i)25-s + 27-s − 6i·29-s − 4·31-s + 4i·33-s + (−4 + 8i)35-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + (0.894 + 0.447i)5-s + 1.51i·7-s + 0.333·9-s + 1.20i·11-s + (0.516 + 0.258i)15-s + 0.970i·17-s + 0.872i·21-s − 0.834i·23-s + (0.600 + 0.800i)25-s + 0.192·27-s − 1.11i·29-s − 0.718·31-s + 0.696i·33-s + (−0.676 + 1.35i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.316 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.316 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.691925082\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.691925082\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + (-2 - i)T \) |
| good | 7 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 - 4iT - 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 4iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 4iT - 47T^{2} \) |
| 53 | \( 1 + 12T + 53T^{2} \) |
| 59 | \( 1 - 4iT - 59T^{2} \) |
| 61 | \( 1 + 2iT - 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 8iT - 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 8iT - 97T^{2} \) |
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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
−8.790165115090069543544324857368, −8.053489724833257968419918576678, −7.30212945367935585218786793098, −6.24848184896888642789081534520, −5.99125627803011138061045023709, −4.96060550476014776285030158685, −4.15436479906600123882530211764, −2.90110785206999153107689811766, −2.32285434847346042853356653305, −1.67853602402807342262235977792,
0.70526333358588687860869285987, 1.51878371058921089288147181231, 2.79061836263945101001896086968, 3.54249169822309023329639140240, 4.40843713773104647927832779843, 5.24134973955034089485771777697, 6.04433641931386524583642277522, 6.91750512104070747468753812056, 7.54209363257644188226591206895, 8.276555353635208334176176248068
