L(s) = 1 | + (0.262 − 1.71i)3-s + (−2.76 − 2.76i)5-s − 7-s + (−2.86 − 0.899i)9-s + (3.92 − 3.92i)11-s + (−1.26 − 1.26i)13-s + (−5.45 + 4.00i)15-s − 7.10i·17-s + (0.652 − 0.652i)19-s + (−0.262 + 1.71i)21-s + 3.98i·23-s + 10.2i·25-s + (−2.29 + 4.66i)27-s + (−0.280 + 0.280i)29-s + 2.51i·31-s + ⋯ |
L(s) = 1 | + (0.151 − 0.988i)3-s + (−1.23 − 1.23i)5-s − 0.377·7-s + (−0.954 − 0.299i)9-s + (1.18 − 1.18i)11-s + (−0.351 − 0.351i)13-s + (−1.40 + 1.03i)15-s − 1.72i·17-s + (0.149 − 0.149i)19-s + (−0.0573 + 0.373i)21-s + 0.830i·23-s + 2.05i·25-s + (−0.440 + 0.897i)27-s + (−0.0520 + 0.0520i)29-s + 0.452i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.764 - 0.644i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.764 - 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8380894231\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8380894231\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.262 + 1.71i)T \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 + (2.76 + 2.76i)T + 5iT^{2} \) |
| 11 | \( 1 + (-3.92 + 3.92i)T - 11iT^{2} \) |
| 13 | \( 1 + (1.26 + 1.26i)T + 13iT^{2} \) |
| 17 | \( 1 + 7.10iT - 17T^{2} \) |
| 19 | \( 1 + (-0.652 + 0.652i)T - 19iT^{2} \) |
| 23 | \( 1 - 3.98iT - 23T^{2} \) |
| 29 | \( 1 + (0.280 - 0.280i)T - 29iT^{2} \) |
| 31 | \( 1 - 2.51iT - 31T^{2} \) |
| 37 | \( 1 + (-2.94 + 2.94i)T - 37iT^{2} \) |
| 41 | \( 1 + 5.39T + 41T^{2} \) |
| 43 | \( 1 + (-1.15 - 1.15i)T + 43iT^{2} \) |
| 47 | \( 1 + 5.00T + 47T^{2} \) |
| 53 | \( 1 + (-6.25 - 6.25i)T + 53iT^{2} \) |
| 59 | \( 1 + (1.75 - 1.75i)T - 59iT^{2} \) |
| 61 | \( 1 + (2.02 + 2.02i)T + 61iT^{2} \) |
| 67 | \( 1 + (-0.986 + 0.986i)T - 67iT^{2} \) |
| 71 | \( 1 - 5.74iT - 71T^{2} \) |
| 73 | \( 1 + 0.913iT - 73T^{2} \) |
| 79 | \( 1 + 3.77iT - 79T^{2} \) |
| 83 | \( 1 + (-9.11 - 9.11i)T + 83iT^{2} \) |
| 89 | \( 1 + 2.10T + 89T^{2} \) |
| 97 | \( 1 - 11.0T + 97T^{2} \) |
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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
−8.991911681328702337627583248320, −8.297530368569480456580123582290, −7.52658590994083900056059799432, −6.88421634837941509771183787338, −5.77750166669925218000387698999, −4.93296845176690923825997037841, −3.75398700547158496545759588259, −2.97758439703052891262864977289, −1.20627662908876132484546086671, −0.38106134599954180876627238675,
2.21409846186844937626122794023, 3.45394685648861324278360426443, 3.96320910505528974220651022918, 4.67662429744323936295995918325, 6.22684835712055893547313741916, 6.76468090305889159639331806938, 7.72193427189609373511510230011, 8.490564740350814511621950044437, 9.437677852466945430600056527811, 10.22757315334237684644609132420