L(s) = 1 | + (−1.5 − 0.866i)3-s + 5-s + (−2.5 − 0.866i)7-s + (1.5 + 2.59i)9-s − 3·11-s + (−0.5 − 0.866i)13-s + (−1.5 − 0.866i)15-s + (−1.5 − 2.59i)17-s + (−2.5 + 4.33i)19-s + (3 + 3.46i)21-s + 23-s − 4·25-s − 5.19i·27-s + (−4.5 + 7.79i)29-s + (−2 + 3.46i)31-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.499i)3-s + 0.447·5-s + (−0.944 − 0.327i)7-s + (0.5 + 0.866i)9-s − 0.904·11-s + (−0.138 − 0.240i)13-s + (−0.387 − 0.223i)15-s + (−0.363 − 0.630i)17-s + (−0.573 + 0.993i)19-s + (0.654 + 0.755i)21-s + 0.208·23-s − 0.800·25-s − 0.999i·27-s + (−0.835 + 1.44i)29-s + (−0.359 + 0.622i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.902 - 0.430i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.902 - 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]
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 | \( 1 + (1.5 + 0.866i)T \) |
| 7 | \( 1 + (2.5 + 0.866i)T \) |
good | 5 | \( 1 - T + 5T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 + (0.5 + 0.866i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.5 - 4.33i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - T + 23T^{2} \) |
| 29 | \( 1 + (4.5 - 7.79i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2.5 - 4.33i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (3.5 + 6.06i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.5 - 2.59i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (4 + 6.92i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.5 + 7.79i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6 - 10.3i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + (-6.5 - 11.2i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-6.5 + 11.2i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-4.5 + 7.79i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-8.5 + 14.7i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.34836231675990158122457594446, −9.881707600079495059170256717032, −8.572544378603291298885114852962, −7.41804678390037314637467274775, −6.71919860045487598999978081608, −5.74977613807558241122157740740, −4.97504431922675873586174400249, −3.41768171506507303580053096014, −1.91450305252295693519187425443, 0,
2.33548856750677211636087214042, 3.79088052368507204221615201809, 4.93342821910264251036151756285, 5.94798816023374966874477759482, 6.50394113790698887435063618559, 7.74334515630327033354897986943, 9.148442425590475418981790287437, 9.663956534569453107209949519381, 10.56511679564692285714522832656