L(s) = 1 | + (0.5 − 0.866i)3-s + (−1.5 + 0.866i)5-s + (−0.499 − 0.866i)9-s + (1.5 + 0.866i)11-s + 1.73i·15-s + (−3 − 1.73i)17-s + (−1 − 1.73i)19-s + (−1 + 1.73i)25-s − 0.999·27-s + 9·29-s + (2.5 − 4.33i)31-s + (1.5 − 0.866i)33-s + (−5 − 8.66i)37-s − 10.3i·41-s − 3.46i·43-s + ⋯ |
L(s) = 1 | + (0.288 − 0.499i)3-s + (−0.670 + 0.387i)5-s + (−0.166 − 0.288i)9-s + (0.452 + 0.261i)11-s + 0.447i·15-s + (−0.727 − 0.420i)17-s + (−0.229 − 0.397i)19-s + (−0.200 + 0.346i)25-s − 0.192·27-s + 1.67·29-s + (0.449 − 0.777i)31-s + (0.261 − 0.150i)33-s + (−0.821 − 1.42i)37-s − 1.62i·41-s − 0.528i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.316138050\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.316138050\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (1.5 - 0.866i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.5 - 0.866i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + (3 + 1.73i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 9T + 29T^{2} \) |
| 31 | \( 1 + (-2.5 + 4.33i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (5 + 8.66i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 10.3iT - 41T^{2} \) |
| 43 | \( 1 + 3.46iT - 43T^{2} \) |
| 47 | \( 1 + (-6 - 10.3i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.5 + 7.79i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (4.5 - 7.79i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (12 + 6.92i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 13.8iT - 71T^{2} \) |
| 73 | \( 1 + (-6 - 3.46i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.5 + 2.59i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 3T + 83T^{2} \) |
| 89 | \( 1 + (-3 + 1.73i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 19.0iT - 97T^{2} \) |
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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
−8.855426110933115167239576857527, −7.88454516855995810949953044389, −7.24605705839342267284631427107, −6.68485477892148032298159891672, −5.79074193678024563218607245976, −4.62522903507757384733851379372, −3.88160556942459854174710190481, −2.89028599935860106670883267044, −1.96589583707472775622036054929, −0.47187175054487656799691651162,
1.19258533009139570742679586053, 2.60934635459595210091445547858, 3.57790410341216103499082643008, 4.38124431498908982707228474431, 4.95070527027467724661806520015, 6.17263184419040992947263615415, 6.78901410403490639059555040879, 7.944839739836297320405585783842, 8.472143172246000693225470748079, 8.952111334548776175456974282711