L(s) = 1 | + (1.05 + 0.942i)2-s + (0.450 − 0.780i)3-s + (0.224 + 1.98i)4-s + (−0.866 + 0.5i)5-s + (1.21 − 0.398i)6-s + (2.29 − 1.30i)7-s + (−1.63 + 2.30i)8-s + (1.09 + 1.89i)9-s + (−1.38 − 0.288i)10-s + (−3.24 − 1.87i)11-s + (1.65 + 0.720i)12-s − 2.41i·13-s + (3.65 + 0.786i)14-s + 0.901i·15-s + (−3.89 + 0.893i)16-s + (−0.505 − 0.291i)17-s + ⋯ |
L(s) = 1 | + (0.745 + 0.666i)2-s + (0.260 − 0.450i)3-s + (0.112 + 0.993i)4-s + (−0.387 + 0.223i)5-s + (0.494 − 0.162i)6-s + (0.869 − 0.494i)7-s + (−0.578 + 0.815i)8-s + (0.364 + 0.631i)9-s + (−0.437 − 0.0912i)10-s + (−0.977 − 0.564i)11-s + (0.477 + 0.207i)12-s − 0.671i·13-s + (0.977 + 0.210i)14-s + 0.232i·15-s + (−0.974 + 0.223i)16-s + (−0.122 − 0.0707i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 - 0.686i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.727 - 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.51627 + 0.602703i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.51627 + 0.602703i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.05 - 0.942i)T \) |
| 5 | \( 1 + (0.866 - 0.5i)T \) |
| 7 | \( 1 + (-2.29 + 1.30i)T \) |
good | 3 | \( 1 + (-0.450 + 0.780i)T + (-1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (3.24 + 1.87i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 2.41iT - 13T^{2} \) |
| 17 | \( 1 + (0.505 + 0.291i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.07 + 5.33i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.73 - 2.15i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 0.435T + 29T^{2} \) |
| 31 | \( 1 + (1.26 - 2.19i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.65 - 9.78i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 7.35iT - 41T^{2} \) |
| 43 | \( 1 + 5.80iT - 43T^{2} \) |
| 47 | \( 1 + (-5.78 - 10.0i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.55 + 2.69i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (1.73 - 3.00i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (8.99 - 5.19i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-8.52 - 4.92i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 9.96iT - 71T^{2} \) |
| 73 | \( 1 + (-8.48 - 4.89i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.397 - 0.229i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 2.59T + 83T^{2} \) |
| 89 | \( 1 + (8.55 - 4.94i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 4.54iT - 97T^{2} \) |
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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
−13.47513048592132123555270679449, −12.63590317335905382777674322269, −11.34022943142358581334421184072, −10.55304466097550970140774310331, −8.471211872099136082982058530948, −7.82309515260218287084229605712, −7.00413715997886162678685351831, −5.43671977429896200813599271191, −4.33655550833731760856653921653, −2.65107506927194138372006946687,
2.11382729163852091343236472100, 3.92058722558684412646066651561, 4.78218277491513923428527885602, 6.14727836368313211976366631084, 7.82180742586486276777287909524, 9.112660710245761400245143288739, 10.13228689678029649169606485277, 11.11907057057630546404580334982, 12.19426007219403964968425835023, 12.74760199313875191910957336817