| L(s) = 1 | + 2.73i·2-s − 5.50·4-s − 2.84i·5-s + 3.93i·7-s − 9.58i·8-s + 7.78·10-s − i·11-s + (1.78 − 3.13i)13-s − 10.7·14-s + 15.2·16-s + 3.81·17-s + 2.94i·19-s + 15.6i·20-s + 2.73·22-s − 1.89·23-s + ⋯ |
| L(s) = 1 | + 1.93i·2-s − 2.75·4-s − 1.27i·5-s + 1.48i·7-s − 3.38i·8-s + 2.46·10-s − 0.301i·11-s + (0.494 − 0.868i)13-s − 2.87·14-s + 3.81·16-s + 0.926·17-s + 0.674i·19-s + 3.49i·20-s + 0.583·22-s − 0.395·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.868 - 0.494i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.868 - 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.300052921\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.300052921\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + iT \) |
| 13 | \( 1 + (-1.78 + 3.13i)T \) |
| good | 2 | \( 1 - 2.73iT - 2T^{2} \) |
| 5 | \( 1 + 2.84iT - 5T^{2} \) |
| 7 | \( 1 - 3.93iT - 7T^{2} \) |
| 17 | \( 1 - 3.81T + 17T^{2} \) |
| 19 | \( 1 - 2.94iT - 19T^{2} \) |
| 23 | \( 1 + 1.89T + 23T^{2} \) |
| 29 | \( 1 - 2.09T + 29T^{2} \) |
| 31 | \( 1 - 6.16iT - 31T^{2} \) |
| 37 | \( 1 - 8.34iT - 37T^{2} \) |
| 41 | \( 1 - 6.35iT - 41T^{2} \) |
| 43 | \( 1 - 11.7T + 43T^{2} \) |
| 47 | \( 1 + 5.31iT - 47T^{2} \) |
| 53 | \( 1 - 2.37T + 53T^{2} \) |
| 59 | \( 1 - 5.38iT - 59T^{2} \) |
| 61 | \( 1 + 2.34T + 61T^{2} \) |
| 67 | \( 1 + 10.4iT - 67T^{2} \) |
| 71 | \( 1 - 10.0iT - 71T^{2} \) |
| 73 | \( 1 + 15.1iT - 73T^{2} \) |
| 79 | \( 1 - 1.57T + 79T^{2} \) |
| 83 | \( 1 - 10.6iT - 83T^{2} \) |
| 89 | \( 1 + 3.23iT - 89T^{2} \) |
| 97 | \( 1 - 17.7iT - 97T^{2} \) |
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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
−9.432818418928330620675953161723, −8.848632204973460319485145584249, −8.228132243630768289717982591948, −7.87125568269895299528844067183, −6.49629995379660474626302939476, −5.71069688423370543830820384180, −5.38259786043232729825541594567, −4.54708966083346867313626350082, −3.29521325893619840769118889146, −1.05268806966957204364975076946,
0.73357932470839572211726494725, 2.03290070968914788647464606754, 3.02007314817985855139060699061, 3.93346427412586827052340504243, 4.34982763015917038665597707666, 5.79548630369004937283661279967, 7.06743399291015371401037817672, 7.73982789519971235952125300039, 8.954260828028972814701877635888, 9.775424832677834901152509149871
