| L(s) = 1 | + (−1 − 2i)5-s − 3·11-s − 5i·13-s − i·17-s − 6·19-s − 9i·23-s + (−3 + 4i)25-s + 29-s − 3·31-s + 6i·37-s + 10·41-s + 9i·43-s + 3i·47-s + 7·49-s + 4i·53-s + ⋯ |
| L(s) = 1 | + (−0.447 − 0.894i)5-s − 0.904·11-s − 1.38i·13-s − 0.242i·17-s − 1.37·19-s − 1.87i·23-s + (−0.600 + 0.800i)25-s + 0.185·29-s − 0.538·31-s + 0.986i·37-s + 1.56·41-s + 1.37i·43-s + 0.437i·47-s + 49-s + 0.549i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + (1 + 2i)T \) |
| good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 + 5iT - 13T^{2} \) |
| 17 | \( 1 + iT - 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 + 9iT - 23T^{2} \) |
| 29 | \( 1 - T + 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 - 9iT - 43T^{2} \) |
| 47 | \( 1 - 3iT - 47T^{2} \) |
| 53 | \( 1 - 4iT - 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 - 9T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 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
−8.004273851844292328934312814288, −7.39462507204383834711528946443, −6.26386417076812809265381507056, −5.68352779358598727389890020794, −4.68868703272931563452990976850, −4.38706513160056980566551081999, −3.11439438803123958940884370455, −2.39483813375047245275032599233, −0.973999443062880107012682626265, 0,
1.84417271437197806202096183312, 2.52818277803571159480371174162, 3.68281938755459980697431255584, 4.13379644009758040713330215748, 5.20930388047662605508862598711, 6.04647686921345974675507844558, 6.74392564294349125400154323280, 7.46591261321200709405576274905, 7.926173921401790536647631764066
