| L(s) = 1 | − 2i·5-s − 2·7-s − 4i·11-s + 2i·13-s + 4·17-s − 4i·19-s + 8·23-s + 25-s + 6i·29-s − 6·31-s + 4i·35-s + 2i·37-s − 12·41-s − 12i·43-s − 8·47-s + ⋯ |
| L(s) = 1 | − 0.894i·5-s − 0.755·7-s − 1.20i·11-s + 0.554i·13-s + 0.970·17-s − 0.917i·19-s + 1.66·23-s + 0.200·25-s + 1.11i·29-s − 1.07·31-s + 0.676i·35-s + 0.328i·37-s − 1.87·41-s − 1.82i·43-s − 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.128267099\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.128267099\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 2iT - 5T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 + 4iT - 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 - 8T + 23T^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 12T + 41T^{2} \) |
| 43 | \( 1 + 12iT - 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 8iT - 59T^{2} \) |
| 61 | \( 1 + 10iT - 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 + 14T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 - 8T + 89T^{2} \) |
| 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
−8.876823183834166789100568286911, −8.152036909016030893130358617456, −6.98078916625179322064920091345, −6.54042905847488682980164784222, −5.21667449640218402278133474906, −5.09522196778896633967211852956, −3.57356561707363237351944766725, −3.11604440093850998158291949664, −1.53562119823161585881403749417, −0.39876046015446341899127553863,
1.46846804517886797743116627437, 2.83041310730183062747333376547, 3.32731163603403322155229686646, 4.45085927168261632939298814040, 5.45701462269163133544264688803, 6.25496000786573372800596195797, 7.09322719693834748955918835130, 7.50878479422072028238821596199, 8.492439858865446723316560468248, 9.542589884284167678380497047633
