| L(s) = 1 | + (0.669 + 0.743i)2-s + (−0.913 − 0.406i)3-s + (−0.104 + 0.994i)4-s + (−2.21 − 0.469i)5-s + (−0.309 − 0.951i)6-s + (2.39 + 1.12i)7-s + (−0.809 + 0.587i)8-s + (0.669 + 0.743i)9-s + (−1.13 − 1.95i)10-s + (2.03 + 2.61i)11-s + (0.5 − 0.866i)12-s + (−0.0900 + 0.276i)13-s + (0.761 + 2.53i)14-s + (1.82 + 1.32i)15-s + (−0.978 − 0.207i)16-s + (−4.15 + 4.61i)17-s + ⋯ |
| L(s) = 1 | + (0.473 + 0.525i)2-s + (−0.527 − 0.234i)3-s + (−0.0522 + 0.497i)4-s + (−0.988 − 0.210i)5-s + (−0.126 − 0.388i)6-s + (0.904 + 0.426i)7-s + (−0.286 + 0.207i)8-s + (0.223 + 0.247i)9-s + (−0.357 − 0.618i)10-s + (0.614 + 0.789i)11-s + (0.144 − 0.249i)12-s + (−0.0249 + 0.0768i)13-s + (0.203 + 0.677i)14-s + (0.472 + 0.343i)15-s + (−0.244 − 0.0519i)16-s + (−1.00 + 1.11i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.323 - 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.323 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.698252 + 0.976373i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.698252 + 0.976373i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.669 - 0.743i)T \) |
| 3 | \( 1 + (0.913 + 0.406i)T \) |
| 7 | \( 1 + (-2.39 - 1.12i)T \) |
| 11 | \( 1 + (-2.03 - 2.61i)T \) |
| good | 5 | \( 1 + (2.21 + 0.469i)T + (4.56 + 2.03i)T^{2} \) |
| 13 | \( 1 + (0.0900 - 0.276i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (4.15 - 4.61i)T + (-1.77 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-0.231 - 2.20i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (3.94 - 6.84i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-5.32 - 3.86i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (0.121 - 0.0259i)T + (28.3 - 12.6i)T^{2} \) |
| 37 | \( 1 + (-5.23 + 2.32i)T + (24.7 - 27.4i)T^{2} \) |
| 41 | \( 1 + (-1.99 + 1.45i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 7.07T + 43T^{2} \) |
| 47 | \( 1 + (0.612 + 5.82i)T + (-45.9 + 9.77i)T^{2} \) |
| 53 | \( 1 + (4.07 - 0.867i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (-1.29 + 12.2i)T + (-57.7 - 12.2i)T^{2} \) |
| 61 | \( 1 + (-3.00 - 0.637i)T + (55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (3.28 + 5.69i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (4.38 + 13.4i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (0.769 - 7.32i)T + (-71.4 - 15.1i)T^{2} \) |
| 79 | \( 1 + (-4.22 - 4.68i)T + (-8.25 + 78.5i)T^{2} \) |
| 83 | \( 1 + (0.346 + 1.06i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (8.72 - 15.1i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.42 + 10.5i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−11.59340184924040897658977930079, −10.75142715902914695171211767700, −9.349740488366833813980409352249, −8.245964025856182169719066142040, −7.69221017360909726540268086589, −6.65702124329541763667504731525, −5.65976916553472578034816587993, −4.56265147196142033240060260725, −3.89188569322950698928852651394, −1.84318896179642668265392665364,
0.71140720456754537713729060341, 2.71957231499540800502679512757, 4.22996059655214751102963364196, 4.53275840742227179342931896783, 5.96904429205061671008495043117, 6.97855525363939218210305828315, 8.054773595838563857992374450775, 9.034920163208498627347858383832, 10.25697576922082651088034068553, 11.14800744565572128911404911619
