L(s) = 1 | + (0.505 + 1.65i)3-s + (−2.00 + 2.00i)5-s − 7-s + (−2.48 + 1.67i)9-s + (−2.67 − 2.67i)11-s + (4.14 − 4.14i)13-s + (−4.34 − 2.31i)15-s − 3.45i·17-s + (−5.27 − 5.27i)19-s + (−0.505 − 1.65i)21-s + 2.11i·23-s − 3.06i·25-s + (−4.03 − 3.27i)27-s + (0.808 + 0.808i)29-s + 7.56i·31-s + ⋯ |
L(s) = 1 | + (0.291 + 0.956i)3-s + (−0.897 + 0.897i)5-s − 0.377·7-s + (−0.829 + 0.558i)9-s + (−0.806 − 0.806i)11-s + (1.14 − 1.14i)13-s + (−1.12 − 0.596i)15-s − 0.838i·17-s + (−1.21 − 1.21i)19-s + (−0.110 − 0.361i)21-s + 0.441i·23-s − 0.612i·25-s + (−0.776 − 0.630i)27-s + (0.150 + 0.150i)29-s + 1.35i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.360 + 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.360 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5567999210\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5567999210\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.505 - 1.65i)T \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 + (2.00 - 2.00i)T - 5iT^{2} \) |
| 11 | \( 1 + (2.67 + 2.67i)T + 11iT^{2} \) |
| 13 | \( 1 + (-4.14 + 4.14i)T - 13iT^{2} \) |
| 17 | \( 1 + 3.45iT - 17T^{2} \) |
| 19 | \( 1 + (5.27 + 5.27i)T + 19iT^{2} \) |
| 23 | \( 1 - 2.11iT - 23T^{2} \) |
| 29 | \( 1 + (-0.808 - 0.808i)T + 29iT^{2} \) |
| 31 | \( 1 - 7.56iT - 31T^{2} \) |
| 37 | \( 1 + (1.06 + 1.06i)T + 37iT^{2} \) |
| 41 | \( 1 - 11.1T + 41T^{2} \) |
| 43 | \( 1 + (-5.32 + 5.32i)T - 43iT^{2} \) |
| 47 | \( 1 + 6.19T + 47T^{2} \) |
| 53 | \( 1 + (0.414 - 0.414i)T - 53iT^{2} \) |
| 59 | \( 1 + (7.26 + 7.26i)T + 59iT^{2} \) |
| 61 | \( 1 + (1.06 - 1.06i)T - 61iT^{2} \) |
| 67 | \( 1 + (4.81 + 4.81i)T + 67iT^{2} \) |
| 71 | \( 1 - 1.83iT - 71T^{2} \) |
| 73 | \( 1 - 0.150iT - 73T^{2} \) |
| 79 | \( 1 + 4.73iT - 79T^{2} \) |
| 83 | \( 1 + (-1.94 + 1.94i)T - 83iT^{2} \) |
| 89 | \( 1 + 11.7T + 89T^{2} \) |
| 97 | \( 1 + 15.6T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.419061182982082950605153441872, −8.616393453849697935307848154559, −7.999215196928304148220295416028, −7.10618980849816311756719643190, −6.06637655487364721535519854983, −5.19845777682606908184012922077, −4.11911352212970181200197444691, −3.15690662974139321514630825217, −2.84322875332042116888422901539, −0.22747600106912395203757553262,
1.34093120502412119828857792512, 2.40705607359747622256923060370, 3.90973728398834884047907196396, 4.37719957246953770794371239655, 5.90591943886710570931280893272, 6.42866286494063464647403629663, 7.57084472306492931028403735520, 8.123529996096057039613831118547, 8.701767328848664075251243806389, 9.533395824890454728582551595810