| L(s) = 1 | − i·3-s − i·5-s − 9-s + 2i·13-s − 15-s + 6·17-s − 4i·19-s − 8·23-s − 25-s + i·27-s − 2i·29-s + 4·31-s − 10i·37-s + 2·39-s − 2·41-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.447i·5-s − 0.333·9-s + 0.554i·13-s − 0.258·15-s + 1.45·17-s − 0.917i·19-s − 1.66·23-s − 0.200·25-s + 0.192i·27-s − 0.371i·29-s + 0.718·31-s − 1.64i·37-s + 0.320·39-s − 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.341664838\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.341664838\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + iT \) |
| good | 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + 8T + 23T^{2} \) |
| 29 | \( 1 + 2iT - 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 + 8iT - 59T^{2} \) |
| 61 | \( 1 + 2iT - 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 14T + 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−7.965743589608975679233923328142, −7.71604590658453268096394262956, −6.70192636153825363612406782316, −6.02953872046730374709736573289, −5.30441820394794900917805333909, −4.40475011416131343569874803981, −3.55120584020870056508697459347, −2.46609638879247422239295151276, −1.56169011630807567410406283985, −0.40653967383218646315047504887,
1.26802235711140916350607979965, 2.55338153179805031541196666305, 3.41670308392441788508564955804, 4.02502835484179707680358310849, 5.08152324919443367659255227682, 5.78690743690175185560878556349, 6.37962823724393061817821724156, 7.47041099164964697221049263451, 8.035570985380482134476749926426, 8.650999426864013446260208181150
