L(s) = 1 | + 4.33i·5-s + (1.65 + 2.06i)7-s − 3.79i·11-s + 2.82i·13-s − 4.33i·17-s − 2.54·19-s − 5.64i·23-s − 13.8·25-s − 9.50·29-s + 1.84·31-s + (−8.96 + 7.16i)35-s − 5.11·37-s + 1.32i·41-s + 2.47i·43-s − 12.2·47-s + ⋯ |
L(s) = 1 | + 1.93i·5-s + (0.624 + 0.781i)7-s − 1.14i·11-s + 0.784i·13-s − 1.05i·17-s − 0.582·19-s − 1.17i·23-s − 2.76·25-s − 1.76·29-s + 0.331·31-s + (−1.51 + 1.21i)35-s − 0.841·37-s + 0.206i·41-s + 0.377i·43-s − 1.78·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.781 + 0.624i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.781 + 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4370003231\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4370003231\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.65 - 2.06i)T \) |
good | 5 | \( 1 - 4.33iT - 5T^{2} \) |
| 11 | \( 1 + 3.79iT - 11T^{2} \) |
| 13 | \( 1 - 2.82iT - 13T^{2} \) |
| 17 | \( 1 + 4.33iT - 17T^{2} \) |
| 19 | \( 1 + 2.54T + 19T^{2} \) |
| 23 | \( 1 + 5.64iT - 23T^{2} \) |
| 29 | \( 1 + 9.50T + 29T^{2} \) |
| 31 | \( 1 - 1.84T + 31T^{2} \) |
| 37 | \( 1 + 5.11T + 37T^{2} \) |
| 41 | \( 1 - 1.32iT - 41T^{2} \) |
| 43 | \( 1 - 2.47iT - 43T^{2} \) |
| 47 | \( 1 + 12.2T + 47T^{2} \) |
| 53 | \( 1 - 1.23T + 53T^{2} \) |
| 59 | \( 1 + 1.65T + 59T^{2} \) |
| 61 | \( 1 - 3.50iT - 61T^{2} \) |
| 67 | \( 1 - 11.1iT - 67T^{2} \) |
| 71 | \( 1 - 3.03iT - 71T^{2} \) |
| 73 | \( 1 + 3.01iT - 73T^{2} \) |
| 79 | \( 1 + 10.1iT - 79T^{2} \) |
| 83 | \( 1 - 3.04T + 83T^{2} \) |
| 89 | \( 1 + 6.53iT - 89T^{2} \) |
| 97 | \( 1 + 6.33iT - 97T^{2} \) |
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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
−8.846819616914460868030814962392, −8.107525395600058916336354828568, −7.33936172708728812297723866748, −6.62905994329698544783621515466, −6.11996027397149856365840319354, −5.31478776578501955634409661886, −4.23679480402258698937083564828, −3.28128901830693339618305091416, −2.64864288892857368652685498841, −1.86883695930919853426939847607,
0.11551815526513963941706193184, 1.46066185824358006984345042888, 1.84652889593333714789830182542, 3.72277538846010816763995755559, 4.17906864329983589001411269008, 5.14195904188806857460337021991, 5.35025182185396314477523093677, 6.52000701278497944276069348714, 7.64404016116395724778095686826, 7.924510125604735724090986808715