L(s) = 1 | + 2.73i·2-s − 1.15·3-s − 5.47·4-s − 1.87i·5-s − 3.16i·6-s − 9.50i·8-s − 1.66·9-s + 5.13·10-s + 2.29i·11-s + 6.32·12-s + (0.574 − 3.55i)13-s + 2.17i·15-s + 15.0·16-s + 6.07·17-s − 4.54i·18-s + 5.15i·19-s + ⋯ |
L(s) = 1 | + 1.93i·2-s − 0.667·3-s − 2.73·4-s − 0.839i·5-s − 1.29i·6-s − 3.36i·8-s − 0.554·9-s + 1.62·10-s + 0.693i·11-s + 1.82·12-s + (0.159 − 0.987i)13-s + 0.560i·15-s + 3.75·16-s + 1.47·17-s − 1.07i·18-s + 1.18i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.159 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.159 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.637963 + 0.543294i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.637963 + 0.543294i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (-0.574 + 3.55i)T \) |
good | 2 | \( 1 - 2.73iT - 2T^{2} \) |
| 3 | \( 1 + 1.15T + 3T^{2} \) |
| 5 | \( 1 + 1.87iT - 5T^{2} \) |
| 11 | \( 1 - 2.29iT - 11T^{2} \) |
| 17 | \( 1 - 6.07T + 17T^{2} \) |
| 19 | \( 1 - 5.15iT - 19T^{2} \) |
| 23 | \( 1 + 4.41T + 23T^{2} \) |
| 29 | \( 1 - 7.50T + 29T^{2} \) |
| 31 | \( 1 + 4.33iT - 31T^{2} \) |
| 37 | \( 1 + 3.16iT - 37T^{2} \) |
| 41 | \( 1 + 2.45iT - 41T^{2} \) |
| 43 | \( 1 + 2.17T + 43T^{2} \) |
| 47 | \( 1 + 8.90iT - 47T^{2} \) |
| 53 | \( 1 - 8.10T + 53T^{2} \) |
| 59 | \( 1 + 7.13iT - 59T^{2} \) |
| 61 | \( 1 - 8.00T + 61T^{2} \) |
| 67 | \( 1 - 1.15iT - 67T^{2} \) |
| 71 | \( 1 - 5.11iT - 71T^{2} \) |
| 73 | \( 1 + 1.96iT - 73T^{2} \) |
| 79 | \( 1 - 3.81T + 79T^{2} \) |
| 83 | \( 1 + 2.49iT - 83T^{2} \) |
| 89 | \( 1 + 10.4iT - 89T^{2} \) |
| 97 | \( 1 - 2.51iT - 97T^{2} \) |
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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
−10.29066648869050193299171646099, −9.775833736533140278996837123933, −8.493046846287920259638967582829, −8.169233880583607707906417042066, −7.21533323753488757464986102070, −6.06540395063971917865312966778, −5.54808771991531748870101421809, −4.88734952082295181356137531481, −3.71577362885389210265891286890, −0.69807622465968078829108985231,
1.00361730339868902955540741875, 2.63552976307937293773089309113, 3.31572766332886116019163232882, 4.54024499440299996302027636674, 5.52732000047735720067099375231, 6.61793539134826405978394964294, 8.185502627387466853057757160883, 8.969879806267961687319503461256, 9.965659475867636585504962958163, 10.62898546016706751797468200541