L(s) = 1 | − 2.68·5-s + 4.15i·7-s − 4.35·11-s + (3.53 + 0.726i)13-s − 5.87·17-s + 5.71·19-s + 3.62·23-s + 2.23·25-s + 3.08i·29-s + 9.28i·31-s − 11.1i·35-s − 2.69·37-s + 11.1i·41-s − 3.80i·43-s + 4.91i·47-s + ⋯ |
L(s) = 1 | − 1.20·5-s + 1.56i·7-s − 1.31·11-s + (0.979 + 0.201i)13-s − 1.42·17-s + 1.31·19-s + 0.756·23-s + 0.447·25-s + 0.573i·29-s + 1.66i·31-s − 1.88i·35-s − 0.443·37-s + 1.73i·41-s − 0.580i·43-s + 0.716i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.738 + 0.673i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.738 + 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2094679016\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2094679016\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + (-3.53 - 0.726i)T \) |
good | 5 | \( 1 + 2.68T + 5T^{2} \) |
| 7 | \( 1 - 4.15iT - 7T^{2} \) |
| 11 | \( 1 + 4.35T + 11T^{2} \) |
| 17 | \( 1 + 5.87T + 17T^{2} \) |
| 19 | \( 1 - 5.71T + 19T^{2} \) |
| 23 | \( 1 - 3.62T + 23T^{2} \) |
| 29 | \( 1 - 3.08iT - 29T^{2} \) |
| 31 | \( 1 - 9.28iT - 31T^{2} \) |
| 37 | \( 1 + 2.69T + 37T^{2} \) |
| 41 | \( 1 - 11.1iT - 41T^{2} \) |
| 43 | \( 1 + 3.80iT - 43T^{2} \) |
| 47 | \( 1 - 4.91iT - 47T^{2} \) |
| 53 | \( 1 + 1.17iT - 53T^{2} \) |
| 59 | \( 1 + 2.29T + 59T^{2} \) |
| 61 | \( 1 + 7.05iT - 61T^{2} \) |
| 67 | \( 1 + 10.0T + 67T^{2} \) |
| 71 | \( 1 - 2.08iT - 71T^{2} \) |
| 73 | \( 1 + 13.4iT - 73T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 - 9.73T + 83T^{2} \) |
| 89 | \( 1 + 12.1iT - 89T^{2} \) |
| 97 | \( 1 + 5.13iT - 97T^{2} \) |
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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
−8.770518757142874778078576532516, −8.362922797505197170806553257733, −7.59865968219541157705024896399, −6.81888813703862925513589920051, −5.94171217204846311460896877797, −5.11795617286128924667176366192, −4.58480209998910864261029330580, −3.26531473917593154878393422419, −2.89498397162330211630852529622, −1.61674064223814282620648175016,
0.07500527494715604688295289499, 0.964018436896238674879896646475, 2.52509188019800680677701282083, 3.60789519853199344109995103363, 4.05471347782454822569411054505, 4.84622531472751442601475059680, 5.78630272683835508259394967337, 6.87936630574805624582300260576, 7.43495403919853057234364166428, 7.87465601407067729129408992117