L(s) = 1 | + (0.780 + 1.35i)3-s + (−0.5 − 0.866i)5-s − 4.56·7-s + (0.280 − 0.486i)9-s − 11-s + (1 − 1.73i)13-s + (0.780 − 1.35i)15-s + (1.56 + 2.70i)17-s + (2.5 + 3.57i)19-s + (−3.56 − 6.16i)21-s + (3.84 − 6.65i)23-s + (−0.499 + 0.866i)25-s + 5.56·27-s + (−1 + 1.73i)29-s + 6.24·31-s + ⋯ |
L(s) = 1 | + (0.450 + 0.780i)3-s + (−0.223 − 0.387i)5-s − 1.72·7-s + (0.0935 − 0.162i)9-s − 0.301·11-s + (0.277 − 0.480i)13-s + (0.201 − 0.349i)15-s + (0.378 + 0.655i)17-s + (0.573 + 0.819i)19-s + (−0.777 − 1.34i)21-s + (0.801 − 1.38i)23-s + (−0.0999 + 0.173i)25-s + 1.07·27-s + (−0.185 + 0.321i)29-s + 1.12·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.125i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.529398856\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.529398856\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 19 | \( 1 + (-2.5 - 3.57i)T \) |
good | 3 | \( 1 + (-0.780 - 1.35i)T + (-1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + 4.56T + 7T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 + (-1 + 1.73i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.56 - 2.70i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-3.84 + 6.65i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1 - 1.73i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 6.24T + 31T^{2} \) |
| 37 | \( 1 - 7.68T + 37T^{2} \) |
| 41 | \( 1 + (1.06 + 1.83i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2.56 + 4.43i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-5.56 + 9.63i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.71 + 2.97i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.21 - 9.03i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.56 + 2.70i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.78 + 11.7i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (0.123 + 0.213i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (5.34 + 9.25i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (1.43 + 2.49i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 9.80T + 83T^{2} \) |
| 89 | \( 1 + (4.84 - 8.38i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.34 - 9.25i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.469484838859560201140787403196, −8.803134173703534074947836349402, −8.076761251084299496102462408824, −6.93266955945248097279587910866, −6.21512572708738176895092897895, −5.28773818838591727975644997677, −4.11672522964619612908464678211, −3.48999676238643270143082302585, −2.72562282604721477956386391929, −0.73057802392998758100911751123,
1.02203008894528612553823276662, 2.69748328528249269537214095340, 3.04564060973029732995280668302, 4.28307944301420078982453664634, 5.54679775886287613671985205919, 6.50040431031559685465898789018, 7.11644916572033181370267255196, 7.64036268112074911375512460403, 8.678159407537844613516222537366, 9.625127205776343467536295192768