| L(s) = 1 | + 1.35·3-s + 2.23·5-s + 12.4i·7-s − 7.15·9-s + (7.23 + 8.28i)11-s − 1.28i·13-s + 3.03·15-s + 3.33i·17-s + 1.88i·19-s + 16.8i·21-s − 32.3·23-s + 5.00·25-s − 21.9·27-s − 27.9i·29-s − 16.5·31-s + ⋯ |
| L(s) = 1 | + 0.452·3-s + 0.447·5-s + 1.77i·7-s − 0.795·9-s + (0.658 + 0.752i)11-s − 0.0984i·13-s + 0.202·15-s + 0.195i·17-s + 0.0992i·19-s + 0.804i·21-s − 1.40·23-s + 0.200·25-s − 0.812·27-s − 0.962i·29-s − 0.532·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.658 - 0.752i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.658 - 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.568719461\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.568719461\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 2.23T \) |
| 11 | \( 1 + (-7.23 - 8.28i)T \) |
| good | 3 | \( 1 - 1.35T + 9T^{2} \) |
| 7 | \( 1 - 12.4iT - 49T^{2} \) |
| 13 | \( 1 + 1.28iT - 169T^{2} \) |
| 17 | \( 1 - 3.33iT - 289T^{2} \) |
| 19 | \( 1 - 1.88iT - 361T^{2} \) |
| 23 | \( 1 + 32.3T + 529T^{2} \) |
| 29 | \( 1 + 27.9iT - 841T^{2} \) |
| 31 | \( 1 + 16.5T + 961T^{2} \) |
| 37 | \( 1 + 22.4T + 1.36e3T^{2} \) |
| 41 | \( 1 - 52.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 15.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 87.6T + 2.20e3T^{2} \) |
| 53 | \( 1 - 74.2T + 2.80e3T^{2} \) |
| 59 | \( 1 - 26.8T + 3.48e3T^{2} \) |
| 61 | \( 1 - 47.4iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 79.2T + 4.48e3T^{2} \) |
| 71 | \( 1 - 74.9T + 5.04e3T^{2} \) |
| 73 | \( 1 + 64.0iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 151. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 151. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 127.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 88.1T + 9.40e3T^{2} \) |
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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
−9.846295948939086560243434636846, −9.467382993343378238810575345174, −8.532088198403774591248039537643, −8.073003329632306951861393552270, −6.62629554687197118713077835644, −5.87466858015511334616001779281, −5.16440988747391055541179045496, −3.78227723331065384497756434373, −2.55208308179361631086020213608, −1.91417148535558647514893524750,
0.45047255545888444479670962289, 1.81559215652423149754039217583, 3.33356032051949083235110749430, 3.93772724642397647516762134632, 5.21218705648221710860702466780, 6.28909675084330528548213608205, 7.07174508236284818165621494879, 7.998837290281867015950975644971, 8.775223383919063442212542794050, 9.654821061388281428729890226770
