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 | \( 1 + (-5.34 + 9.25i)T + (-48.5 - 84.0i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.625127205776343467536295192768, −8.678159407537844613516222537366, −7.64036268112074911375512460403, −7.11644916572033181370267255196, −6.50040431031559685465898789018, −5.54679775886287613671985205919, −4.28307944301420078982453664634, −3.04564060973029732995280668302, −2.69748328528249269537214095340, −1.02203008894528612553823276662,
0.73057802392998758100911751123, 2.72562282604721477956386391929, 3.48999676238643270143082302585, 4.11672522964619612908464678211, 5.28773818838591727975644997677, 6.21512572708738176895092897895, 6.93266955945248097279587910866, 8.076761251084299496102462408824, 8.803134173703534074947836349402, 9.469484838859560201140787403196