L(s) = 1 | − 0.585i·5-s + 0.828i·7-s − 2.82·11-s + 2.82·13-s − 2.58i·17-s − 5.65i·19-s + 6.82·23-s + 4.65·25-s − 3.41i·29-s + 8.82i·31-s + 0.485·35-s + 7.65·37-s − 5.41i·41-s − 1.65i·43-s − 4.48·47-s + ⋯ |
L(s) = 1 | − 0.261i·5-s + 0.313i·7-s − 0.852·11-s + 0.784·13-s − 0.627i·17-s − 1.29i·19-s + 1.42·23-s + 0.931·25-s − 0.634i·29-s + 1.58i·31-s + 0.0820·35-s + 1.25·37-s − 0.845i·41-s − 0.252i·43-s − 0.654·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.586923297\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.586923297\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.585iT - 5T^{2} \) |
| 7 | \( 1 - 0.828iT - 7T^{2} \) |
| 11 | \( 1 + 2.82T + 11T^{2} \) |
| 13 | \( 1 - 2.82T + 13T^{2} \) |
| 17 | \( 1 + 2.58iT - 17T^{2} \) |
| 19 | \( 1 + 5.65iT - 19T^{2} \) |
| 23 | \( 1 - 6.82T + 23T^{2} \) |
| 29 | \( 1 + 3.41iT - 29T^{2} \) |
| 31 | \( 1 - 8.82iT - 31T^{2} \) |
| 37 | \( 1 - 7.65T + 37T^{2} \) |
| 41 | \( 1 + 5.41iT - 41T^{2} \) |
| 43 | \( 1 + 1.65iT - 43T^{2} \) |
| 47 | \( 1 + 4.48T + 47T^{2} \) |
| 53 | \( 1 + 9.07iT - 53T^{2} \) |
| 59 | \( 1 + 13.6T + 59T^{2} \) |
| 61 | \( 1 - 3.65T + 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 - 10.4iT - 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 + 3.75iT - 89T^{2} \) |
| 97 | \( 1 - 2.34T + 97T^{2} \) |
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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
−9.534351442215529866530562301275, −8.924756478024221044963727304835, −8.203220991943803733349745743876, −7.17995062855596276371499875977, −6.45459819917204749709549008344, −5.22603830306411940939075843950, −4.80047392369487570334180443794, −3.34230647187902475315749829755, −2.46025955931186664150547554228, −0.830953813554536544138076068711,
1.19972866375679529908112501987, 2.67244674310645454888764126950, 3.65536204286048010856906490920, 4.67458696876862102829131237348, 5.74917625443140913397069840128, 6.46277004407426450396466057182, 7.53774696270985509759176555930, 8.126275158912990645243560800994, 9.057793105354217262396810583219, 9.962799567892150161201490562542