| L(s) = 1 | + (0.913 + 0.406i)2-s + (1.33 − 1.48i)3-s + (0.669 + 0.743i)4-s + (0.0399 − 0.379i)5-s + (1.82 − 0.813i)6-s + (1.30 − 4.02i)7-s + (0.309 + 0.951i)8-s + (−0.104 − 0.994i)9-s + (0.190 − 0.330i)10-s + (−2.54 + 2.12i)11-s + 2·12-s + (0.193 + 1.84i)13-s + (2.83 − 3.14i)14-s + (−0.511 − 0.567i)15-s + (−0.104 + 0.994i)16-s + (0.562 − 5.35i)17-s + ⋯ |
| L(s) = 1 | + (0.645 + 0.287i)2-s + (0.772 − 0.858i)3-s + (0.334 + 0.371i)4-s + (0.0178 − 0.169i)5-s + (0.745 − 0.332i)6-s + (0.494 − 1.52i)7-s + (0.109 + 0.336i)8-s + (−0.0348 − 0.331i)9-s + (0.0603 − 0.104i)10-s + (−0.767 + 0.641i)11-s + 0.577·12-s + (0.0537 + 0.511i)13-s + (0.757 − 0.841i)14-s + (−0.131 − 0.146i)15-s + (−0.0261 + 0.248i)16-s + (0.136 − 1.29i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.39291 - 0.785929i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.39291 - 0.785929i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.913 - 0.406i)T \) |
| 11 | \( 1 + (2.54 - 2.12i)T \) |
| 19 | \( 1 + (3.06 - 3.09i)T \) |
| good | 3 | \( 1 + (-1.33 + 1.48i)T + (-0.313 - 2.98i)T^{2} \) |
| 5 | \( 1 + (-0.0399 + 0.379i)T + (-4.89 - 1.03i)T^{2} \) |
| 7 | \( 1 + (-1.30 + 4.02i)T + (-5.66 - 4.11i)T^{2} \) |
| 13 | \( 1 + (-0.193 - 1.84i)T + (-12.7 + 2.70i)T^{2} \) |
| 17 | \( 1 + (-0.562 + 5.35i)T + (-16.6 - 3.53i)T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.49 - 1.66i)T + (-3.03 + 28.8i)T^{2} \) |
| 31 | \( 1 + (7.97 - 5.79i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (1.04 - 3.21i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-6.65 + 7.39i)T + (-4.28 - 40.7i)T^{2} \) |
| 43 | \( 1 + (-5.35 - 9.27i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (0.978 + 0.207i)T + (42.9 + 19.1i)T^{2} \) |
| 53 | \( 1 + (0.707 + 6.72i)T + (-51.8 + 11.0i)T^{2} \) |
| 59 | \( 1 + (8.51 - 1.81i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (6.52 - 2.90i)T + (40.8 - 45.3i)T^{2} \) |
| 67 | \( 1 + (6.16 - 10.6i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.661 + 6.29i)T + (-69.4 - 14.7i)T^{2} \) |
| 73 | \( 1 + (-10.9 + 2.33i)T + (66.6 - 29.6i)T^{2} \) |
| 79 | \( 1 + (-3.38 - 1.50i)T + (52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (9.28 + 6.74i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (7.35 - 12.7i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (14.7 + 6.58i)T + (64.9 + 72.0i)T^{2} \) |
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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
−11.09745944509121799459422445083, −10.42010937070001391381780953012, −9.059456630235787565581049410442, −7.960126319650647600702900876855, −7.34373349368307077984342685173, −6.80629393882822794882944409035, −5.12515449803109428223548708853, −4.27862508974589053668301998898, −2.90555911058353405547765069364, −1.54349182161311252416799071506,
2.30971197117600677923579672738, 3.12647739969252389659094789290, 4.30754562254805971078730545996, 5.42886772620657333793188264447, 6.16763838387565922123538568070, 7.898494776695209463991757786903, 8.715977747076625594798849323288, 9.389832924648024682078409277867, 10.65737863821094041920240496072, 11.05137851351390408293016844521
