| L(s) = 1 | − i·3-s + 2i·5-s + 2i·7-s − 9-s − 2·13-s + 2·15-s + (−1 − 4i)17-s − 4·19-s + 2·21-s + 2i·23-s + 25-s + i·27-s + 2i·29-s + 2i·31-s − 4·35-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.894i·5-s + 0.755i·7-s − 0.333·9-s − 0.554·13-s + 0.516·15-s + (−0.242 − 0.970i)17-s − 0.917·19-s + 0.436·21-s + 0.417i·23-s + 0.200·25-s + 0.192i·27-s + 0.371i·29-s + 0.359i·31-s − 0.676·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 + 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3264 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 17 | \( 1 + (1 + 4i)T \) |
| good | 5 | \( 1 - 2iT - 5T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 2iT - 23T^{2} \) |
| 29 | \( 1 - 2iT - 29T^{2} \) |
| 31 | \( 1 - 2iT - 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 6iT - 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 - 2iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 10iT - 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 4iT - 97T^{2} \) |
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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
−8.308214659363619700251168385294, −7.34927671293210359354178302810, −6.91196014220385073134643908580, −6.16862100960228138437264922416, −5.36373660397584487843703308909, −4.50696747772029112664654524817, −3.20349456927637199665081096295, −2.63384933237430905026090025816, −1.71970836998179909189495326566, 0,
1.33236530530530222750183972939, 2.53177668592740285158076870180, 3.75195544275812733848703153186, 4.39562714406982276701382240793, 4.95037755790314960174178662396, 5.93011284460378139820141700956, 6.69344832447220363833996743583, 7.62681910073560329533321596098, 8.428602581108460320200220546494
