L(s) = 1 | + (−1.66 + 0.468i)3-s + 3.33·5-s + (1.56 − 2.13i)7-s + (2.56 − 1.56i)9-s + 4i·11-s − 5.20i·13-s + (−5.56 + 1.56i)15-s − 1.87·17-s − 0.936i·19-s + (−1.60 + 4.29i)21-s − 7.12i·23-s + 6.12·25-s + (−3.54 + 3.80i)27-s + 7.12i·29-s − 2.39i·31-s + ⋯ |
L(s) = 1 | + (−0.962 + 0.270i)3-s + 1.49·5-s + (0.590 − 0.807i)7-s + (0.853 − 0.520i)9-s + 1.20i·11-s − 1.44i·13-s + (−1.43 + 0.403i)15-s − 0.454·17-s − 0.214i·19-s + (−0.350 + 0.936i)21-s − 1.48i·23-s + 1.22·25-s + (−0.681 + 0.731i)27-s + 1.32i·29-s − 0.430i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 + 0.350i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.936 + 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.49931 - 0.270964i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.49931 - 0.270964i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.66 - 0.468i)T \) |
| 7 | \( 1 + (-1.56 + 2.13i)T \) |
good | 5 | \( 1 - 3.33T + 5T^{2} \) |
| 11 | \( 1 - 4iT - 11T^{2} \) |
| 13 | \( 1 + 5.20iT - 13T^{2} \) |
| 17 | \( 1 + 1.87T + 17T^{2} \) |
| 19 | \( 1 + 0.936iT - 19T^{2} \) |
| 23 | \( 1 + 7.12iT - 23T^{2} \) |
| 29 | \( 1 - 7.12iT - 29T^{2} \) |
| 31 | \( 1 + 2.39iT - 31T^{2} \) |
| 37 | \( 1 - 8.24T + 37T^{2} \) |
| 41 | \( 1 + 1.87T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 - 6.67T + 47T^{2} \) |
| 53 | \( 1 - 0.876iT - 53T^{2} \) |
| 59 | \( 1 - 7.08T + 59T^{2} \) |
| 61 | \( 1 - 8.13iT - 61T^{2} \) |
| 67 | \( 1 + 14.2T + 67T^{2} \) |
| 71 | \( 1 + 2.24iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 3.12T + 79T^{2} \) |
| 83 | \( 1 - 6.25T + 83T^{2} \) |
| 89 | \( 1 + 4.79T + 89T^{2} \) |
| 97 | \( 1 + 2.92iT - 97T^{2} \) |
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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
−10.48054008224796769189291743017, −9.915550861898781575039734133819, −8.983242873939117088224197356997, −7.61294085792463192706239509141, −6.77444026724640094778505286184, −5.87056321413901905380938843645, −5.04485547176643110540121209345, −4.27797895483756676419784234497, −2.43917774901478156456105955391, −1.07056919577681234930869176654,
1.45358080890331479416041109268, 2.39070189515421328305050853046, 4.30436387511883994361305225292, 5.50325963451225259257542351512, 5.89780421669818729535245693527, 6.67114044515813026207283068064, 7.918472278322398995180259129875, 9.102798883041358087893421729278, 9.584036900790060210459956506818, 10.72147787706537242614215890113