| L(s) = 1 | − 0.359i·5-s + 4.15i·7-s + 11-s + 1.77·13-s + 2.30i·17-s − 3.49i·19-s − 8.02·23-s + 4.87·25-s − 7.15i·29-s − 10.6i·31-s + 1.49·35-s − 0.323·37-s − 12.6i·41-s − 11.7i·43-s − 4.28·47-s + ⋯ |
| L(s) = 1 | − 0.160i·5-s + 1.56i·7-s + 0.301·11-s + 0.491·13-s + 0.557i·17-s − 0.801i·19-s − 1.67·23-s + 0.974·25-s − 1.32i·29-s − 1.90i·31-s + 0.252·35-s − 0.0532·37-s − 1.97i·41-s − 1.79i·43-s − 0.624·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.736176035\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.736176035\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + 0.359iT - 5T^{2} \) |
| 7 | \( 1 - 4.15iT - 7T^{2} \) |
| 13 | \( 1 - 1.77T + 13T^{2} \) |
| 17 | \( 1 - 2.30iT - 17T^{2} \) |
| 19 | \( 1 + 3.49iT - 19T^{2} \) |
| 23 | \( 1 + 8.02T + 23T^{2} \) |
| 29 | \( 1 + 7.15iT - 29T^{2} \) |
| 31 | \( 1 + 10.6iT - 31T^{2} \) |
| 37 | \( 1 + 0.323T + 37T^{2} \) |
| 41 | \( 1 + 12.6iT - 41T^{2} \) |
| 43 | \( 1 + 11.7iT - 43T^{2} \) |
| 47 | \( 1 + 4.28T + 47T^{2} \) |
| 53 | \( 1 - 12.1iT - 53T^{2} \) |
| 59 | \( 1 + 0.755T + 59T^{2} \) |
| 61 | \( 1 - 13.8T + 61T^{2} \) |
| 67 | \( 1 + 6.03iT - 67T^{2} \) |
| 71 | \( 1 - 3.14T + 71T^{2} \) |
| 73 | \( 1 + 1.39T + 73T^{2} \) |
| 79 | \( 1 - 9.80iT - 79T^{2} \) |
| 83 | \( 1 - 1.07T + 83T^{2} \) |
| 89 | \( 1 + 5.97iT - 89T^{2} \) |
| 97 | \( 1 - 14.0T + 97T^{2} \) |
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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
−8.127051707473970977868183495386, −7.31049457486329443500857721404, −6.29132321881802770016282977171, −5.90935747147147687904341243058, −5.27391566590162099352766725723, −4.26749537448372353279501535123, −3.62362681587991602036057073636, −2.37020122709485771967277504750, −2.07104611200558152217726760903, −0.50779767433980920515412579004,
0.965996366080803961313240828523, 1.67865047153266758342495684331, 3.15180560039211854299471101454, 3.59511047858315342106223943839, 4.49390733166821384465044329152, 5.07266425636753248913147947079, 6.22821937064683431678776688519, 6.71009234637906940636776869774, 7.32643205839788349209265571663, 8.113454470429720275384109325186
