L(s) = 1 | − 3i·3-s + 15.6·5-s + (10.8 + 15.0i)7-s − 9·9-s + 41.3·11-s + 8.02·13-s − 47.0i·15-s − 14.2i·17-s + 10.2i·19-s + (45.0 − 32.5i)21-s − 203. i·23-s + 121.·25-s + 27i·27-s − 82.6i·29-s − 156.·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + 1.40·5-s + (0.585 + 0.810i)7-s − 0.333·9-s + 1.13·11-s + 0.171·13-s − 0.810i·15-s − 0.202i·17-s + 0.123i·19-s + (0.468 − 0.337i)21-s − 1.84i·23-s + 0.969·25-s + 0.192i·27-s − 0.529i·29-s − 0.905·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.775 + 0.631i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.775 + 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(3.455911324\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.455911324\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 3iT \) |
| 7 | \( 1 + (-10.8 - 15.0i)T \) |
good | 5 | \( 1 - 15.6T + 125T^{2} \) |
| 11 | \( 1 - 41.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 8.02T + 2.19e3T^{2} \) |
| 17 | \( 1 + 14.2iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 10.2iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 203. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 82.6iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 156.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 106. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 315. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 154.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 474.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 266. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 425. iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 8.30T + 2.26e5T^{2} \) |
| 67 | \( 1 - 820.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 109. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 53.8iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 1.17e3iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 507. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 1.06e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 573. iT - 9.12e5T^{2} \) |
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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
−8.929929803400927974065846593209, −8.684266836044143522986122055369, −7.43225804641765267321087292721, −6.50347256136270227391385629043, −5.92907283005503446166512598467, −5.21331772371770665817725580246, −4.02328539691396421282545939052, −2.48489112628561117288005295703, −1.98141952726712361724509106553, −0.879066000493068619142196452798,
1.14549412639454458265780365998, 1.89210774623465044352414027704, 3.33958031795393944455269544911, 4.19559311432910441876427946319, 5.19868056949981225087347089802, 5.90635337307055899109080925437, 6.78141354991349141200987204997, 7.68207902386253784991162333508, 8.797922612878768122969447789111, 9.470877909358310317073919872148