| L(s) = 1 | − i·2-s − 4-s − 3.56i·5-s − i·7-s + i·8-s − 3.56·10-s − 5.56i·11-s + (3.56 − 0.561i)13-s − 14-s + 16-s + 6.68·17-s − 1.56i·19-s + 3.56i·20-s − 5.56·22-s + 6.68·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s − 1.59i·5-s − 0.377i·7-s + 0.353i·8-s − 1.12·10-s − 1.67i·11-s + (0.987 − 0.155i)13-s − 0.267·14-s + 0.250·16-s + 1.62·17-s − 0.358i·19-s + 0.796i·20-s − 1.18·22-s + 1.39·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.155i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 + 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.712294717\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.712294717\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
| 13 | \( 1 + (-3.56 + 0.561i)T \) |
| good | 5 | \( 1 + 3.56iT - 5T^{2} \) |
| 11 | \( 1 + 5.56iT - 11T^{2} \) |
| 17 | \( 1 - 6.68T + 17T^{2} \) |
| 19 | \( 1 + 1.56iT - 19T^{2} \) |
| 23 | \( 1 - 6.68T + 23T^{2} \) |
| 29 | \( 1 + 1.56T + 29T^{2} \) |
| 31 | \( 1 + 6.24iT - 31T^{2} \) |
| 37 | \( 1 - 10.6iT - 37T^{2} \) |
| 41 | \( 1 + 4iT - 41T^{2} \) |
| 43 | \( 1 + 6.43T + 43T^{2} \) |
| 47 | \( 1 - 10.2iT - 47T^{2} \) |
| 53 | \( 1 + 4.87T + 53T^{2} \) |
| 59 | \( 1 + 4.24iT - 59T^{2} \) |
| 61 | \( 1 + 1.56T + 61T^{2} \) |
| 67 | \( 1 + 1.12iT - 67T^{2} \) |
| 71 | \( 1 + 9.36iT - 71T^{2} \) |
| 73 | \( 1 - 11.5iT - 73T^{2} \) |
| 79 | \( 1 - 16T + 79T^{2} \) |
| 83 | \( 1 - 2iT - 83T^{2} \) |
| 89 | \( 1 - 8iT - 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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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
−9.087215346202593666223669527962, −8.266109254897207544455661951689, −7.928263629217223323925607049098, −6.34096176114589322306752118090, −5.48794289661882500812260238102, −4.87276974728336136352407586228, −3.75222765895104211986372594665, −3.09636240823409532044314619111, −1.26616881102446534843715848896, −0.795665259039742726776993912761,
1.71621497689666381853842177433, 3.03367840443466072854087526775, 3.76967506519886699183327857410, 5.01724340342661590528111489322, 5.84198170489369896371537570362, 6.75897295586517752051462585975, 7.19361048267812400131178866690, 7.903424079163973091979333935954, 8.950358820177132473187542115741, 9.796957885979981185360019714641
