| L(s) = 1 | + (−2.51 + 0.887i)2-s + (−0.372 + 0.927i)3-s + (3.96 − 3.20i)4-s + (−2.24 − 3.64i)5-s + (0.113 − 2.66i)6-s + (0.412 − 3.21i)7-s + (−4.33 + 7.03i)8-s + (−0.721 − 0.691i)9-s + (8.88 + 7.17i)10-s + (0.0936 − 2.20i)11-s + (1.49 + 4.87i)12-s + (−1.85 − 4.60i)13-s + (1.82 + 8.45i)14-s + (4.22 − 0.723i)15-s + (2.48 − 11.5i)16-s + (0.187 − 4.41i)17-s + ⋯ |
| L(s) = 1 | + (−1.77 + 0.627i)2-s + (−0.215 + 0.535i)3-s + (1.98 − 1.60i)4-s + (−1.00 − 1.63i)5-s + (0.0461 − 1.08i)6-s + (0.155 − 1.21i)7-s + (−1.53 + 2.48i)8-s + (−0.240 − 0.230i)9-s + (2.80 + 2.26i)10-s + (0.0282 − 0.665i)11-s + (0.431 + 1.40i)12-s + (−0.513 − 1.27i)13-s + (0.486 + 2.25i)14-s + (1.09 − 0.186i)15-s + (0.622 − 2.88i)16-s + (0.0454 − 1.06i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 669 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.919 + 0.393i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 669 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.919 + 0.393i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0608022 - 0.296801i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0608022 - 0.296801i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.372 - 0.927i)T \) |
| 223 | \( 1 + (-12.8 - 7.64i)T \) |
| good | 2 | \( 1 + (2.51 - 0.887i)T + (1.55 - 1.25i)T^{2} \) |
| 5 | \( 1 + (2.24 + 3.64i)T + (-2.25 + 4.46i)T^{2} \) |
| 7 | \( 1 + (-0.412 + 3.21i)T + (-6.77 - 1.76i)T^{2} \) |
| 11 | \( 1 + (-0.0936 + 2.20i)T + (-10.9 - 0.932i)T^{2} \) |
| 13 | \( 1 + (1.85 + 4.60i)T + (-9.38 + 8.99i)T^{2} \) |
| 17 | \( 1 + (-0.187 + 4.41i)T + (-16.9 - 1.44i)T^{2} \) |
| 19 | \( 1 + (-4.32 + 1.12i)T + (16.5 - 9.26i)T^{2} \) |
| 23 | \( 1 + (-0.0604 - 0.0981i)T + (-10.3 + 20.5i)T^{2} \) |
| 29 | \( 1 + (1.60 - 7.43i)T + (-26.4 - 11.9i)T^{2} \) |
| 31 | \( 1 + (6.61 - 0.563i)T + (30.5 - 5.23i)T^{2} \) |
| 37 | \( 1 + (-0.240 - 5.66i)T + (-36.8 + 3.13i)T^{2} \) |
| 41 | \( 1 + (-9.50 + 6.42i)T + (15.2 - 38.0i)T^{2} \) |
| 43 | \( 1 + (2.86 + 5.68i)T + (-25.5 + 34.5i)T^{2} \) |
| 47 | \( 1 + (3.16 + 0.823i)T + (41.0 + 22.9i)T^{2} \) |
| 53 | \( 1 + (-0.675 + 0.768i)T + (-6.73 - 52.5i)T^{2} \) |
| 59 | \( 1 + (-0.281 + 0.918i)T + (-48.8 - 33.0i)T^{2} \) |
| 61 | \( 1 + (-4.52 - 11.2i)T + (-44.0 + 42.2i)T^{2} \) |
| 67 | \( 1 + (-7.76 + 6.26i)T + (14.1 - 65.4i)T^{2} \) |
| 71 | \( 1 + (0.125 - 0.410i)T + (-58.8 - 39.7i)T^{2} \) |
| 73 | \( 1 + (8.51 + 4.75i)T + (38.2 + 62.1i)T^{2} \) |
| 79 | \( 1 + (-6.06 + 4.09i)T + (29.4 - 73.3i)T^{2} \) |
| 83 | \( 1 + (0.529 - 1.05i)T + (-49.3 - 66.7i)T^{2} \) |
| 89 | \( 1 + (-6.52 + 10.6i)T + (-40.0 - 79.4i)T^{2} \) |
| 97 | \( 1 + (0.0974 - 0.158i)T + (-43.6 - 86.6i)T^{2} \) |
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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
−9.903112595854743064876077916749, −9.101363211096670987984421251066, −8.522275958474538223615240408756, −7.51614999064112985633741534893, −7.32043434765042613439929808484, −5.54420721499553672654044181688, −4.95589894186549353178080193595, −3.46047879574185903620425440715, −1.03114763427211098237064426585, −0.36726147142276003856668891398,
1.96151074002061852786156994752, 2.62421831635583655601347812190, 3.86854405190138849980140827290, 6.10670690652877423013221937627, 6.91185387549506879711070717515, 7.62129653351934915257444714865, 8.153882240912490313017774552792, 9.328341140126667404587896186293, 9.929691239520071322925678041248, 11.07897360074545376555272458523
