L(s) = 1 | + (−1 − 1.41i)3-s − i·5-s − 0.828i·7-s + (−1.00 + 2.82i)9-s + 0.828·11-s − 6.82·13-s + (−1.41 + i)15-s + 4.82i·17-s − 6i·19-s + (−1.17 + 0.828i)21-s − 4.82·23-s − 25-s + (5.00 − 1.41i)27-s + 6i·29-s − 2i·31-s + ⋯ |
L(s) = 1 | + (−0.577 − 0.816i)3-s − 0.447i·5-s − 0.313i·7-s + (−0.333 + 0.942i)9-s + 0.249·11-s − 1.89·13-s + (−0.365 + 0.258i)15-s + 1.17i·17-s − 1.37i·19-s + (−0.255 + 0.180i)21-s − 1.00·23-s − 0.200·25-s + (0.962 − 0.272i)27-s + 1.11i·29-s − 0.359i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
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 | \( 1 + (1 + 1.41i)T \) |
| 5 | \( 1 + iT \) |
good | 7 | \( 1 + 0.828iT - 7T^{2} \) |
| 11 | \( 1 - 0.828T + 11T^{2} \) |
| 13 | \( 1 + 6.82T + 13T^{2} \) |
| 17 | \( 1 - 4.82iT - 17T^{2} \) |
| 19 | \( 1 + 6iT - 19T^{2} \) |
| 23 | \( 1 + 4.82T + 23T^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 + 2iT - 31T^{2} \) |
| 37 | \( 1 + 1.17T + 37T^{2} \) |
| 41 | \( 1 - 1.65iT - 41T^{2} \) |
| 43 | \( 1 - 6.82iT - 43T^{2} \) |
| 47 | \( 1 + 8.82T + 47T^{2} \) |
| 53 | \( 1 - 3.65iT - 53T^{2} \) |
| 59 | \( 1 + 7.17T + 59T^{2} \) |
| 61 | \( 1 + 9.31T + 61T^{2} \) |
| 67 | \( 1 - 12.4iT - 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 + 6iT - 79T^{2} \) |
| 83 | \( 1 + 17.3T + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−9.538879696764750128817154855959, −8.516016957104936525993530452311, −7.64564359947976449219769865975, −6.99672014787326310009687654019, −6.10152579156254975570106931642, −5.09221389519400601324645609303, −4.37098589681666745752731822547, −2.71773310562645337857709105076, −1.54905764480344454072129526475, 0,
2.26394288354445839833496159260, 3.42670708120199287503316310364, 4.50060447417012674780479392208, 5.32211957938522430639226692980, 6.17071591565130486877509420643, 7.14093559646151513839879572791, 8.012191755092277339944217254228, 9.242921476799586251957047367611, 9.875975661817253618125152200745