| L(s) = 1 | + 1.32·5-s − 0.339i·7-s − 0.576i·11-s − 2.03i·13-s − 6.46·17-s + (2.76 + 3.36i)19-s + 1.23i·23-s − 3.24·25-s + 2.19i·29-s − 4.72·31-s − 0.449i·35-s + 6.47i·37-s − 4.29i·41-s − 10.4i·43-s − 11.2i·47-s + ⋯ |
| L(s) = 1 | + 0.591·5-s − 0.128i·7-s − 0.173i·11-s − 0.565i·13-s − 1.56·17-s + (0.634 + 0.773i)19-s + 0.257i·23-s − 0.649·25-s + 0.407i·29-s − 0.848·31-s − 0.0759i·35-s + 1.06i·37-s − 0.670i·41-s − 1.59i·43-s − 1.64i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.01116583971\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01116583971\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-2.76 - 3.36i)T \) |
| good | 5 | \( 1 - 1.32T + 5T^{2} \) |
| 7 | \( 1 + 0.339iT - 7T^{2} \) |
| 11 | \( 1 + 0.576iT - 11T^{2} \) |
| 13 | \( 1 + 2.03iT - 13T^{2} \) |
| 17 | \( 1 + 6.46T + 17T^{2} \) |
| 23 | \( 1 - 1.23iT - 23T^{2} \) |
| 29 | \( 1 - 2.19iT - 29T^{2} \) |
| 31 | \( 1 + 4.72T + 31T^{2} \) |
| 37 | \( 1 - 6.47iT - 37T^{2} \) |
| 41 | \( 1 + 4.29iT - 41T^{2} \) |
| 43 | \( 1 + 10.4iT - 43T^{2} \) |
| 47 | \( 1 + 11.2iT - 47T^{2} \) |
| 53 | \( 1 - 10.9iT - 53T^{2} \) |
| 59 | \( 1 + 11.6T + 59T^{2} \) |
| 61 | \( 1 + 8.98T + 61T^{2} \) |
| 67 | \( 1 + 8.33T + 67T^{2} \) |
| 71 | \( 1 + 12.2T + 71T^{2} \) |
| 73 | \( 1 + 1.82T + 73T^{2} \) |
| 79 | \( 1 + 8.98T + 79T^{2} \) |
| 83 | \( 1 + 6.79iT - 83T^{2} \) |
| 89 | \( 1 - 12.8iT - 89T^{2} \) |
| 97 | \( 1 - 17.6iT - 97T^{2} \) |
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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
−7.62153705771523639146465674490, −7.21398369459079978706513777872, −6.21222017796451956550664149051, −5.72735962446160576326694640691, −4.96028636561195993075121776728, −4.04953841930929657835571533104, −3.25923940730309563837277512190, −2.26467646470647182618638804650, −1.44723145954480773770170373535, −0.00264733350710663652747999784,
1.53383863738214157251756083673, 2.31198229317508696131086450595, 3.14059315720256691001741989504, 4.38243301184150254201881201100, 4.69097201476192467681494772256, 5.88165921973109531892595193866, 6.22849787364265269534374941230, 7.15843200445401053525767245304, 7.66652387033187187899126597901, 8.759077081805664691224386459909
