| L(s) = 1 | + (0.669 − 0.743i)2-s + (0.913 − 0.406i)3-s + (−0.104 − 0.994i)4-s + (0.282 − 0.0601i)5-s + (0.309 − 0.951i)6-s + (1.34 + 2.27i)7-s + (−0.809 − 0.587i)8-s + (0.669 − 0.743i)9-s + (0.144 − 0.250i)10-s + (3.27 − 0.532i)11-s + (−0.5 − 0.866i)12-s + (−0.973 − 2.99i)13-s + (2.59 + 0.524i)14-s + (0.233 − 0.170i)15-s + (−0.978 + 0.207i)16-s + (2.65 + 2.95i)17-s + ⋯ |
| L(s) = 1 | + (0.473 − 0.525i)2-s + (0.527 − 0.234i)3-s + (−0.0522 − 0.497i)4-s + (0.126 − 0.0268i)5-s + (0.126 − 0.388i)6-s + (0.508 + 0.860i)7-s + (−0.286 − 0.207i)8-s + (0.223 − 0.247i)9-s + (0.0457 − 0.0792i)10-s + (0.987 − 0.160i)11-s + (−0.144 − 0.249i)12-s + (−0.269 − 0.830i)13-s + (0.693 + 0.140i)14-s + (0.0604 − 0.0438i)15-s + (−0.244 + 0.0519i)16-s + (0.644 + 0.715i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.552 + 0.833i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.552 + 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.01382 - 1.08072i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.01382 - 1.08072i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.34 - 2.27i)T \) |
| 11 | \( 1 + (-3.27 + 0.532i)T \) |
| good | 5 | \( 1 + (-0.282 + 0.0601i)T + (4.56 - 2.03i)T^{2} \) |
| 13 | \( 1 + (0.973 + 2.99i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-2.65 - 2.95i)T + (-1.77 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-0.616 + 5.86i)T + (-18.5 - 3.95i)T^{2} \) |
| 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 | \( 1 + (4.30 + 0.915i)T + (28.3 + 12.6i)T^{2} \) |
| 37 | \( 1 + (-7.01 - 3.12i)T + (24.7 + 27.4i)T^{2} \) |
| 41 | \( 1 + (3.30 + 2.39i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 8.89T + 43T^{2} \) |
| 47 | \( 1 + (0.427 - 4.06i)T + (-45.9 - 9.77i)T^{2} \) |
| 53 | \( 1 + (-5.39 - 1.14i)T + (48.4 + 21.5i)T^{2} \) |
| 59 | \( 1 + (-1.13 - 10.7i)T + (-57.7 + 12.2i)T^{2} \) |
| 61 | \( 1 + (-5.85 + 1.24i)T + (55.7 - 24.8i)T^{2} \) |
| 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 | \( 1 + (0.763 + 7.26i)T + (-71.4 + 15.1i)T^{2} \) |
| 79 | \( 1 + (9.90 - 11.0i)T + (-8.25 - 78.5i)T^{2} \) |
| 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
−11.17831969155745252058211738317, −9.960395880059537049811292203883, −9.160410204542280855335343008342, −8.359621527811637891348339149202, −7.25022041776671003576387899776, −6.00832757720701685025491369147, −5.16241441796182907240293335382, −3.84249114107417210189692162874, −2.74818195005437080838103367060, −1.51446225368230892337279061740,
1.82569587512245668732920118024, 3.62636256332634953695860099690, 4.25320671593650508190611894707, 5.44318511211982160790494413606, 6.66607302876339282318535203641, 7.50417021072377538430058968826, 8.269837690984954968463373143609, 9.459357171042273448334118103998, 10.04695108551103565085681685756, 11.41622683063524937649762694337
