| L(s) = 1 | + 3-s + 0.941i·7-s + 9-s − 4.49i·11-s + 5.55·13-s − 7.55i·17-s + 1.05i·19-s + 0.941i·21-s + 1.05i·23-s + 27-s + 2i·29-s − 3.55·31-s − 4.49i·33-s − 7.43·37-s + 5.55·39-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.355i·7-s + 0.333·9-s − 1.35i·11-s + 1.54·13-s − 1.83i·17-s + 0.242i·19-s + 0.205i·21-s + 0.220i·23-s + 0.192·27-s + 0.371i·29-s − 0.638·31-s − 0.783i·33-s − 1.22·37-s + 0.889·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.606 + 0.794i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.606 + 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.263161558\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.263161558\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 0.941iT - 7T^{2} \) |
| 11 | \( 1 + 4.49iT - 11T^{2} \) |
| 13 | \( 1 - 5.55T + 13T^{2} \) |
| 17 | \( 1 + 7.55iT - 17T^{2} \) |
| 19 | \( 1 - 1.05iT - 19T^{2} \) |
| 23 | \( 1 - 1.05iT - 23T^{2} \) |
| 29 | \( 1 - 2iT - 29T^{2} \) |
| 31 | \( 1 + 3.55T + 31T^{2} \) |
| 37 | \( 1 + 7.43T + 37T^{2} \) |
| 41 | \( 1 + 3.88T + 41T^{2} \) |
| 43 | \( 1 + 1.88T + 43T^{2} \) |
| 47 | \( 1 + 10.0iT - 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 8.49iT - 59T^{2} \) |
| 61 | \( 1 - 8.99iT - 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 12.9T + 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 - 11.5T + 79T^{2} \) |
| 83 | \( 1 - 5.88T + 83T^{2} \) |
| 89 | \( 1 - 4.11T + 89T^{2} \) |
| 97 | \( 1 + 17.1iT - 97T^{2} \) |
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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
−8.738519681500098348332404805979, −8.357130189427668794288567852544, −7.34976509347759265874039864533, −6.57049953000727198155915988785, −5.68303422865833318385383979087, −4.99908492486562611148322788205, −3.58914634324196703489370340258, −3.29770649476703901681287696962, −2.05236991245424410707104141295, −0.75956929801200665649749317828,
1.36528714491591577329387285995, 2.16922650646852252975210900009, 3.58259352866861253422616305858, 4.00636078270904823089089889224, 5.01558600988631020394020957032, 6.14446201947570414896676947317, 6.75390879726032345461048178176, 7.66992977124106132448375118591, 8.318365851761539996503590720279, 8.980833175997487257980876594758
