| L(s) = 1 | − i·3-s − 2i·5-s − 4·7-s − 9-s + 4i·11-s − 2i·13-s − 2·15-s − 6·17-s + 4i·19-s + 4i·21-s + 25-s + i·27-s + 2i·29-s − 4·31-s + 4·33-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.894i·5-s − 1.51·7-s − 0.333·9-s + 1.20i·11-s − 0.554i·13-s − 0.516·15-s − 1.45·17-s + 0.917i·19-s + 0.872i·21-s + 0.200·25-s + 0.192i·27-s + 0.371i·29-s − 0.718·31-s + 0.696·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| good | 5 | \( 1 + 2iT - 5T^{2} \) |
| 7 | \( 1 + 4T + 7T^{2} \) |
| 11 | \( 1 - 4iT - 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 2iT - 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 + 10iT - 53T^{2} \) |
| 59 | \( 1 + 4iT - 59T^{2} \) |
| 61 | \( 1 - 6iT - 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 16T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−9.687819352404895063426047923888, −8.998327639266284776293447036244, −8.109292621001179927364041734470, −7.02636487220597968841948174938, −6.43427622228386769543730958206, −5.34799277350386073726216311847, −4.29370400578789576461250993928, −3.06310481619601238631488333900, −1.73207908456882757664741684628, 0,
2.61280563257824176573263690653, 3.33072250231337258947255629984, 4.32137562834720234205738904062, 5.73167023113259948249081162165, 6.55420735276939294351881900748, 7.06992965408008235742663108340, 8.605948567033838895002997751855, 9.199150600618356363333291423906, 9.992657737159952110923654674533
