| L(s) = 1 | + (1.51 − 0.848i)3-s + i·5-s − 3.02i·7-s + (1.56 − 2.56i)9-s − 1.32·11-s + 5.12·13-s + (0.848 + 1.51i)15-s + 2i·17-s − 1.32i·19-s + (−2.56 − 4.56i)21-s + 0.371·23-s − 25-s + (0.185 − 5.19i)27-s − 3.12i·29-s − 4.71i·31-s + ⋯ |
| L(s) = 1 | + (0.871 − 0.489i)3-s + 0.447i·5-s − 1.14i·7-s + (0.520 − 0.853i)9-s − 0.399·11-s + 1.42·13-s + (0.218 + 0.389i)15-s + 0.485i·17-s − 0.303i·19-s + (−0.558 − 0.995i)21-s + 0.0775·23-s − 0.200·25-s + (0.0357 − 0.999i)27-s − 0.579i·29-s − 0.847i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.489 + 0.871i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.489 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.88974 - 1.10612i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.88974 - 1.10612i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.51 + 0.848i)T \) |
| 5 | \( 1 - iT \) |
| good | 7 | \( 1 + 3.02iT - 7T^{2} \) |
| 11 | \( 1 + 1.32T + 11T^{2} \) |
| 13 | \( 1 - 5.12T + 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 + 1.32iT - 19T^{2} \) |
| 23 | \( 1 - 0.371T + 23T^{2} \) |
| 29 | \( 1 + 3.12iT - 29T^{2} \) |
| 31 | \( 1 + 4.71iT - 31T^{2} \) |
| 37 | \( 1 + 5.12T + 37T^{2} \) |
| 41 | \( 1 + 1.12iT - 41T^{2} \) |
| 43 | \( 1 + 7.73iT - 43T^{2} \) |
| 47 | \( 1 - 3.02T + 47T^{2} \) |
| 53 | \( 1 - 12.2iT - 53T^{2} \) |
| 59 | \( 1 - 14.1T + 59T^{2} \) |
| 61 | \( 1 + 3.12T + 61T^{2} \) |
| 67 | \( 1 - 4.34iT - 67T^{2} \) |
| 71 | \( 1 - 3.39T + 71T^{2} \) |
| 73 | \( 1 - 8.24T + 73T^{2} \) |
| 79 | \( 1 - 8.10iT - 79T^{2} \) |
| 83 | \( 1 + 15.1T + 83T^{2} \) |
| 89 | \( 1 - 10.2iT - 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−9.952385589514246451193770236643, −8.887483911903095481013459423093, −8.180758634309556069049138644552, −7.37030832509953015702063078176, −6.71961595879231862913097590575, −5.76328478490022439757087503540, −4.12326062997423796001799214377, −3.60362251330781105921286427909, −2.36431209826502088299901049207, −1.01909066393524675857853061653,
1.65522069524231165117366213529, 2.85038785527138762740415210921, 3.75024211311188530138845145294, 4.94416816240168934404515667837, 5.63881629234477498565990944576, 6.81311806579418750232078707112, 8.047695209833354272128608989606, 8.597005172014709919104332967020, 9.132780614056953960585258794139, 10.00947850772251084971778040230
