L(s) = 1 | + (−0.5 − 0.866i)2-s + (0.686 + 0.396i)3-s + (−0.499 + 0.866i)4-s + (−1.5 + 1.65i)5-s − 0.792i·6-s + (−3 − 1.73i)7-s + 0.999·8-s + (−1.18 − 2.05i)9-s + (2.18 + 0.469i)10-s + (−0.686 + 0.396i)12-s + (−2.68 + 4.65i)13-s + 3.46i·14-s + (−1.68 + 0.543i)15-s + (−0.5 − 0.866i)16-s + (−2.18 − 3.78i)17-s + (−1.18 + 2.05i)18-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.396 + 0.228i)3-s + (−0.249 + 0.433i)4-s + (−0.670 + 0.741i)5-s − 0.323i·6-s + (−1.13 − 0.654i)7-s + 0.353·8-s + (−0.395 − 0.684i)9-s + (0.691 + 0.148i)10-s + (−0.198 + 0.114i)12-s + (−0.745 + 1.29i)13-s + 0.925i·14-s + (−0.435 + 0.140i)15-s + (−0.125 − 0.216i)16-s + (−0.530 − 0.918i)17-s + (−0.279 + 0.484i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.936 - 0.351i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.936 - 0.351i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 + (1.5 - 1.65i)T \) |
| 37 | \( 1 + (-0.5 - 6.06i)T \) |
good | 3 | \( 1 + (-0.686 - 0.396i)T + (1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + (3 + 1.73i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + (2.68 - 4.65i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (2.18 + 3.78i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 8.74T + 23T^{2} \) |
| 29 | \( 1 - 4.40iT - 29T^{2} \) |
| 31 | \( 1 + 1.08iT - 31T^{2} \) |
| 41 | \( 1 + (2.87 - 4.97i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 - 8.11T + 43T^{2} \) |
| 47 | \( 1 + 1.58iT - 47T^{2} \) |
| 53 | \( 1 + (-3.68 + 2.12i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (4.62 - 2.67i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (6.55 + 3.78i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-10.1 - 5.84i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.37 + 7.57i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 6.92iT - 73T^{2} \) |
| 79 | \( 1 + (-10.1 - 5.84i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (0.255 - 0.147i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (11.1 - 6.45i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 5.11T + 97T^{2} \) |
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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
−10.85436413399536939595985692466, −9.782204625008507401500231957809, −9.425937517448192963360001029391, −8.201838672917920221575809326230, −7.08054401987464178557179478376, −6.43710549241103880379807921398, −4.35166462100043483663066563548, −3.55242703230157424368167477410, −2.55310379476121640911379277657, 0,
2.40446390556957536163841765474, 3.90912450530183781296641046838, 5.34272215297827295424750096398, 6.11740602243252145076170784437, 7.54642146519005609743315884400, 8.124948522730085121833255073320, 8.924221954480204673450960015061, 9.843890948368760744055583870273, 10.81498967256871285936363921432