| L(s) = 1 | − 2i·3-s − 5-s + (2.44 − i)7-s − 9-s + 2·13-s + 2i·15-s − 4.89i·17-s + 2i·19-s + (−2 − 4.89i)21-s + 6i·23-s + 25-s − 4i·27-s − 9.79i·29-s + 9.79·31-s + (−2.44 + i)35-s + ⋯ |
| L(s) = 1 | − 1.15i·3-s − 0.447·5-s + (0.925 − 0.377i)7-s − 0.333·9-s + 0.554·13-s + 0.516i·15-s − 1.18i·17-s + 0.458i·19-s + (−0.436 − 1.06i)21-s + 1.25i·23-s + 0.200·25-s − 0.769i·27-s − 1.81i·29-s + 1.75·31-s + (−0.414 + 0.169i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.387 + 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.387 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.868306238\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.868306238\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + (-2.44 + i)T \) |
| good | 3 | \( 1 + 2iT - 3T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 4.89iT - 17T^{2} \) |
| 19 | \( 1 - 2iT - 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + 9.79iT - 29T^{2} \) |
| 31 | \( 1 - 9.79T + 31T^{2} \) |
| 37 | \( 1 - 4.89iT - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 4.89T + 43T^{2} \) |
| 47 | \( 1 - 4.89T + 47T^{2} \) |
| 53 | \( 1 + 4.89iT - 53T^{2} \) |
| 59 | \( 1 + 6iT - 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 + 14.6T + 67T^{2} \) |
| 71 | \( 1 + 6iT - 71T^{2} \) |
| 73 | \( 1 - 4.89iT - 73T^{2} \) |
| 79 | \( 1 - 10iT - 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 + 9.79iT - 89T^{2} \) |
| 97 | \( 1 + 4.89iT - 97T^{2} \) |
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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
−8.318813020074240120382326271084, −8.043716002026452353823916562155, −7.31248648023376401726140327625, −6.66635086971369195451188758905, −5.75748705926315709846008956760, −4.75656397615886530480565881498, −3.94264044781964025811498683169, −2.72279334321926800202505893963, −1.62660092821968019710734016055, −0.73735364301492171204219814332,
1.32349232062296915615677684455, 2.70343574448862352249166765533, 3.77318123122163781428604160620, 4.45567589124835971058304903796, 5.05321482325895381641460440560, 6.00865218586950210771019641746, 6.95831102234029890747916203864, 7.969997984379079146643434814671, 8.716677837546682141087533176723, 9.013795862175382111223014585276
