| L(s) = 1 | + (0.978 − 0.207i)2-s + (−0.104 − 0.994i)3-s + (0.913 − 0.406i)4-s + (0.775 − 0.861i)5-s + (−0.309 − 0.951i)6-s + (−2.61 − 0.400i)7-s + (0.809 − 0.587i)8-s + (−0.978 + 0.207i)9-s + (0.579 − 1.00i)10-s + (−0.307 − 3.30i)11-s + (−0.5 − 0.866i)12-s + (1.36 − 4.18i)13-s + (−2.64 + 0.152i)14-s + (−0.937 − 0.681i)15-s + (0.669 − 0.743i)16-s + (−1.11 − 0.237i)17-s + ⋯ |
| L(s) = 1 | + (0.691 − 0.147i)2-s + (−0.0603 − 0.574i)3-s + (0.456 − 0.203i)4-s + (0.346 − 0.385i)5-s + (−0.126 − 0.388i)6-s + (−0.988 − 0.151i)7-s + (0.286 − 0.207i)8-s + (−0.326 + 0.0693i)9-s + (0.183 − 0.317i)10-s + (−0.0928 − 0.995i)11-s + (−0.144 − 0.250i)12-s + (0.377 − 1.16i)13-s + (−0.705 + 0.0406i)14-s + (−0.242 − 0.175i)15-s + (0.167 − 0.185i)16-s + (−0.270 − 0.0575i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0764 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0764 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.30928 - 1.41350i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.30928 - 1.41350i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.61 + 0.400i)T \) |
| 11 | \( 1 + (0.307 + 3.30i)T \) |
| good | 5 | \( 1 + (-0.775 + 0.861i)T + (-0.522 - 4.97i)T^{2} \) |
| 13 | \( 1 + (-1.36 + 4.18i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (1.11 + 0.237i)T + (15.5 + 6.91i)T^{2} \) |
| 19 | \( 1 + (-5.66 - 2.52i)T + (12.7 + 14.1i)T^{2} \) |
| 23 | \( 1 + (-0.0167 - 0.0290i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.81 - 2.04i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (5.43 + 6.03i)T + (-3.24 + 30.8i)T^{2} \) |
| 37 | \( 1 + (0.801 - 7.62i)T + (-36.1 - 7.69i)T^{2} \) |
| 41 | \( 1 + (-6.14 + 4.46i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 8.54T + 43T^{2} \) |
| 47 | \( 1 + (-3.64 - 1.62i)T + (31.4 + 34.9i)T^{2} \) |
| 53 | \( 1 + (-8.23 - 9.14i)T + (-5.54 + 52.7i)T^{2} \) |
| 59 | \( 1 + (-9.91 + 4.41i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (1.89 - 2.10i)T + (-6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (4.65 - 8.05i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (0.761 + 2.34i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-7.51 + 3.34i)T + (48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (7.45 - 1.58i)T + (72.1 - 32.1i)T^{2} \) |
| 83 | \( 1 + (-2.60 - 8.01i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (1.06 + 1.83i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (4.23 - 13.0i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−10.94186922539645287952008537294, −10.05573566508874786330739302330, −9.048861144982570639700205064799, −7.951503212659903224486297918731, −6.95773106026363101269671634537, −5.84709991139596219465466278221, −5.43484970087070127454289042026, −3.67831095620500626269282821619, −2.83727508344105925172468261677, −1.01878595651910633271406769497,
2.29535826148328832616705489579, 3.47546487957813834901157734444, 4.50728831301050094944072176075, 5.57256597029320244454393123328, 6.62337598919253593241979997556, 7.19667932867757640793296606834, 8.798373592740634249959564981554, 9.626407956815102483675863153117, 10.32986628067834848603943693054, 11.38540300375393480324660806784
