| L(s) = 1 | − 2i·3-s + (1.73 − 2i)7-s − 9-s + 3.46i·11-s + 3.46·13-s + 2·19-s + (−4 − 3.46i)21-s + 3.46·23-s − 4i·27-s − 6·29-s + 8·31-s + 6.92·33-s + 2i·37-s − 6.92i·39-s + 6.92i·41-s + ⋯ |
| L(s) = 1 | − 1.15i·3-s + (0.654 − 0.755i)7-s − 0.333·9-s + 1.04i·11-s + 0.960·13-s + 0.458·19-s + (−0.872 − 0.755i)21-s + 0.722·23-s − 0.769i·27-s − 1.11·29-s + 1.43·31-s + 1.20·33-s + 0.328i·37-s − 1.10i·39-s + 1.08i·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.270 + 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.270 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.269245614\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.269245614\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1.73 + 2i)T \) |
| good | 3 | \( 1 + 2iT - 3T^{2} \) |
| 11 | \( 1 - 3.46iT - 11T^{2} \) |
| 13 | \( 1 - 3.46T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 3.46T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 6.92iT - 41T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + 3.46iT - 61T^{2} \) |
| 67 | \( 1 - 3.46T + 67T^{2} \) |
| 71 | \( 1 - 3.46iT - 71T^{2} \) |
| 73 | \( 1 - 6.92T + 73T^{2} \) |
| 79 | \( 1 + 3.46iT - 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 + 6.92iT - 89T^{2} \) |
| 97 | \( 1 + 13.8T + 97T^{2} \) |
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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
−8.330549399307722681231136335705, −7.80780374960843514020155622324, −7.13965245575270306833707538037, −6.63022586654358343287509420074, −5.69068363100424391488309447272, −4.66239485731007613560756713642, −3.96510677228507957839373706991, −2.68934046662317822319804883812, −1.60936286217737869562792740320, −0.961316787126073530870675360830,
1.09431156484567893088106457886, 2.50306913118602366773978519977, 3.51083230932756602117896284805, 4.13816354859367061272059039858, 5.19194910119785672526996008420, 5.59578275626963092247047398600, 6.50780912466813092928321764902, 7.63402244191088858756788768442, 8.446351582949086084635521503134, 8.994560466076612409316732591061
