| L(s) = 1 | + (−0.104 − 0.994i)2-s + (0.669 − 0.743i)3-s + (−0.978 + 0.207i)4-s + (1.97 − 0.878i)5-s + (−0.809 − 0.587i)6-s + (0.867 + 2.49i)7-s + (0.309 + 0.951i)8-s + (−0.104 − 0.994i)9-s + (−1.08 − 1.87i)10-s + (2.39 − 2.29i)11-s + (−0.5 + 0.866i)12-s + (0.0379 − 0.0275i)13-s + (2.39 − 1.12i)14-s + (0.667 − 2.05i)15-s + (0.913 − 0.406i)16-s + (0.255 − 2.43i)17-s + ⋯ |
| L(s) = 1 | + (−0.0739 − 0.703i)2-s + (0.386 − 0.429i)3-s + (−0.489 + 0.103i)4-s + (0.882 − 0.392i)5-s + (−0.330 − 0.239i)6-s + (0.327 + 0.944i)7-s + (0.109 + 0.336i)8-s + (−0.0348 − 0.331i)9-s + (−0.341 − 0.591i)10-s + (0.721 − 0.692i)11-s + (−0.144 + 0.249i)12-s + (0.0105 − 0.00764i)13-s + (0.640 − 0.300i)14-s + (0.172 − 0.530i)15-s + (0.228 − 0.101i)16-s + (0.0620 − 0.590i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.194 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.194 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.37307 - 1.12770i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.37307 - 1.12770i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.104 + 0.994i)T \) |
| 3 | \( 1 + (-0.669 + 0.743i)T \) |
| 7 | \( 1 + (-0.867 - 2.49i)T \) |
| 11 | \( 1 + (-2.39 + 2.29i)T \) |
| good | 5 | \( 1 + (-1.97 + 0.878i)T + (3.34 - 3.71i)T^{2} \) |
| 13 | \( 1 + (-0.0379 + 0.0275i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.255 + 2.43i)T + (-16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (-3.96 - 0.842i)T + (17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (0.166 - 0.289i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.528 - 1.62i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (6.74 + 3.00i)T + (20.7 + 23.0i)T^{2} \) |
| 37 | \( 1 + (7.02 + 7.79i)T + (-3.86 + 36.7i)T^{2} \) |
| 41 | \( 1 + (-1.71 - 5.27i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 1.20T + 43T^{2} \) |
| 47 | \( 1 + (-3.16 - 0.672i)T + (42.9 + 19.1i)T^{2} \) |
| 53 | \( 1 + (1.04 + 0.464i)T + (35.4 + 39.3i)T^{2} \) |
| 59 | \( 1 + (-4.34 + 0.922i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (0.689 - 0.307i)T + (40.8 - 45.3i)T^{2} \) |
| 67 | \( 1 + (-5.85 - 10.1i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (0.545 + 0.396i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (9.02 - 1.91i)T + (66.6 - 29.6i)T^{2} \) |
| 79 | \( 1 + (0.0104 + 0.0990i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (-5.09 - 3.70i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (6.90 - 11.9i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (5.16 - 3.75i)T + (29.9 - 92.2i)T^{2} \) |
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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
−11.03767999464239501869809639786, −9.666672118763399109155123945637, −9.188431640872029105330211779407, −8.482384583032030291005663287654, −7.30795415953735219929729323616, −5.89976272136070353416907783132, −5.26070571366639267651812962828, −3.65606412936526904694179136731, −2.43227860034922372018529812807, −1.35888000653501212536667332423,
1.71960058974225559570231903166, 3.50853475791389887125122516278, 4.55066627935227371844438474308, 5.63279743063505443485048628438, 6.76657251956732892359138965425, 7.44347965765188342723712002881, 8.559452721624008613843199511937, 9.527345424187109578707839219279, 10.12144776760024257956317056044, 10.90954011500197539985026649524
