| L(s) = 1 | − 3-s − 2i·5-s − 3·7-s − 2·9-s − 3·11-s − 6i·13-s + 2i·15-s − 2i·17-s − 6i·19-s + 3·21-s + 4i·23-s + 25-s + 5·27-s − 4i·29-s + 3·33-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894i·5-s − 1.13·7-s − 0.666·9-s − 0.904·11-s − 1.66i·13-s + 0.516i·15-s − 0.485i·17-s − 1.37i·19-s + 0.654·21-s + 0.834i·23-s + 0.200·25-s + 0.962·27-s − 0.742i·29-s + 0.522·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.164 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.164 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 37 | \( 1 + (-1 - 6i)T \) |
| good | 3 | \( 1 + T + 3T^{2} \) |
| 5 | \( 1 + 2iT - 5T^{2} \) |
| 7 | \( 1 + 3T + 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 6iT - 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 + 4iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 - 6iT - 43T^{2} \) |
| 47 | \( 1 + 3T + 47T^{2} \) |
| 53 | \( 1 + 9T + 53T^{2} \) |
| 59 | \( 1 - 4iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 - 3T + 71T^{2} \) |
| 73 | \( 1 - 9T + 73T^{2} \) |
| 79 | \( 1 - 6iT - 79T^{2} \) |
| 83 | \( 1 - 9T + 83T^{2} \) |
| 89 | \( 1 - 14iT - 89T^{2} \) |
| 97 | \( 1 + 12iT - 97T^{2} \) |
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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
−8.362034708896369058836038054113, −7.79163653674513405929727054482, −6.73421133728748536837135631159, −5.94550052436284735064388823043, −5.23728753485854125207509354617, −4.75127544913418706924960525672, −3.19754724921036881861349042942, −2.76262106266741339110058683983, −0.827917183712101803789862456877, 0,
1.97391687120051610526195232885, 2.99415137854395561503719447266, 3.73377053221049617452831688812, 4.86539845146700151465527556777, 5.87254616825588994738042919968, 6.44161584788000879906344921298, 6.93480755278313848491866791224, 7.946266237009130432831038258671, 8.842188364743402886253391717201
