| L(s) = 1 | + 2i·7-s − 2i·11-s − 5.29i·19-s + 5.29i·23-s − 6i·29-s − 4·31-s + 10.5·37-s − 10.5·41-s − 10.5·43-s − 5.29i·47-s + 3·49-s + 2·53-s + 10i·59-s + 10.5i·61-s − 10.5·71-s + ⋯ |
| L(s) = 1 | + 0.755i·7-s − 0.603i·11-s − 1.21i·19-s + 1.10i·23-s − 1.11i·29-s − 0.718·31-s + 1.73·37-s − 1.65·41-s − 1.61·43-s − 0.771i·47-s + 0.428·49-s + 0.274·53-s + 1.30i·59-s + 1.35i·61-s − 1.25·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.678 + 0.734i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.678 + 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6609213474\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6609213474\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 5.29iT - 19T^{2} \) |
| 23 | \( 1 - 5.29iT - 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 + 10.5T + 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 + 5.29iT - 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 - 10iT - 59T^{2} \) |
| 61 | \( 1 - 10.5iT - 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 10.5T + 71T^{2} \) |
| 73 | \( 1 + 14iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 + 10.5T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−7.61976647192307859163461806825, −7.03073766503956670914285740817, −6.14020966136358786222065633530, −5.62841233467662731406461839967, −4.91395439197091983631136908019, −4.05071593427363694848708499230, −3.14649904213259377605055347783, −2.48125938381601461449981113301, −1.45952156608427372654918724053, −0.16098090057466365070545768108,
1.19583697725156501334209753969, 2.04106277711974041484944051575, 3.14280230994007865509285076715, 3.87330821380371377864244324413, 4.59923403765107005792553259856, 5.28225863369228703720079043978, 6.20652365437724467115269929692, 6.83714484701256574649601352332, 7.43071461452197338916349838070, 8.187618079130268011606047974001
