| L(s) = 1 | + (−0.978 + 0.207i)2-s + (−0.104 − 0.994i)3-s + (0.913 − 0.406i)4-s + (−0.193 + 0.214i)5-s + (0.309 + 0.951i)6-s + (−1.75 + 1.98i)7-s + (−0.809 + 0.587i)8-s + (−0.978 + 0.207i)9-s + (0.144 − 0.250i)10-s + (−1.17 − 3.10i)11-s + (−0.5 − 0.866i)12-s + (−0.973 + 2.99i)13-s + (1.29 − 2.30i)14-s + (0.233 + 0.170i)15-s + (0.669 − 0.743i)16-s + (−3.88 − 0.825i)17-s + ⋯ |
| L(s) = 1 | + (−0.691 + 0.147i)2-s + (−0.0603 − 0.574i)3-s + (0.456 − 0.203i)4-s + (−0.0865 + 0.0961i)5-s + (0.126 + 0.388i)6-s + (−0.661 + 0.749i)7-s + (−0.286 + 0.207i)8-s + (−0.326 + 0.0693i)9-s + (0.0457 − 0.0792i)10-s + (−0.354 − 0.935i)11-s + (−0.144 − 0.250i)12-s + (−0.269 + 0.830i)13-s + (0.347 − 0.615i)14-s + (0.0604 + 0.0438i)15-s + (0.167 − 0.185i)16-s + (−0.942 − 0.200i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 - 0.369i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.929 - 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0180503 + 0.0942731i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0180503 + 0.0942731i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.978 - 0.207i)T \) |
| 3 | \( 1 + (0.104 + 0.994i)T \) |
| 7 | \( 1 + (1.75 - 1.98i)T \) |
| 11 | \( 1 + (1.17 + 3.10i)T \) |
| good | 5 | \( 1 + (0.193 - 0.214i)T + (-0.522 - 4.97i)T^{2} \) |
| 13 | \( 1 + (0.973 - 2.99i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (3.88 + 0.825i)T + (15.5 + 6.91i)T^{2} \) |
| 19 | \( 1 + (5.39 + 2.40i)T + (12.7 + 14.1i)T^{2} \) |
| 23 | \( 1 + (-0.00853 - 0.0147i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (4.10 + 2.98i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-2.94 - 3.27i)T + (-3.24 + 30.8i)T^{2} \) |
| 37 | \( 1 + (0.802 - 7.63i)T + (-36.1 - 7.69i)T^{2} \) |
| 41 | \( 1 + (3.30 - 2.39i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 8.89T + 43T^{2} \) |
| 47 | \( 1 + (-3.73 - 1.66i)T + (31.4 + 34.9i)T^{2} \) |
| 53 | \( 1 + (3.69 + 4.09i)T + (-5.54 + 52.7i)T^{2} \) |
| 59 | \( 1 + (9.91 - 4.41i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (4.00 - 4.44i)T + (-6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (-5.12 + 8.87i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (4.24 + 13.0i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-6.67 + 2.97i)T + (48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (-14.4 + 3.07i)T + (72.1 - 32.1i)T^{2} \) |
| 83 | \( 1 + (-1.54 - 4.73i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-0.835 - 1.44i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (4.62 - 14.2i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−11.38945230449327800727575808969, −10.61557212844325675822579044481, −9.365287684789097148316943169602, −8.804221871728534942535627225394, −7.925780780843415977531005000580, −6.67174387176999106484545632907, −6.30720527440206483071709382394, −4.96149680254837922497075692898, −3.16340562238804104792030435696, −2.01333696874821024920807456692,
0.06819372579633513102518435849, 2.26503696053534106844035933608, 3.69058192067512201647697402137, 4.68521802516428873916461578759, 6.11791350456044216114841781390, 7.08211887426365906743404688028, 8.037792878592849286313463020618, 8.974696703707590715390662564645, 10.00512694619313277733991638080, 10.38459770328681964728925100207
