| L(s) = 1 | − i·3-s + (−1.58 − 1.57i)5-s + (2.37 − 1.15i)7-s − 9-s + 1.26i·11-s − 6.82·13-s + (−1.57 + 1.58i)15-s + 1.90·17-s − 2.53·19-s + (−1.15 − 2.37i)21-s − 5.50·23-s + (0.0146 + 4.99i)25-s + i·27-s + 6.49·29-s − 5.78·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + (−0.708 − 0.706i)5-s + (0.899 − 0.436i)7-s − 0.333·9-s + 0.380i·11-s − 1.89·13-s + (−0.407 + 0.408i)15-s + 0.461·17-s − 0.581·19-s + (−0.252 − 0.519i)21-s − 1.14·23-s + (0.00292 + 0.999i)25-s + 0.192i·27-s + 1.20·29-s − 1.03·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.655 - 0.754i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.655 - 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1057992978\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1057992978\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (1.58 + 1.57i)T \) |
| 7 | \( 1 + (-2.37 + 1.15i)T \) |
| good | 11 | \( 1 - 1.26iT - 11T^{2} \) |
| 13 | \( 1 + 6.82T + 13T^{2} \) |
| 17 | \( 1 - 1.90T + 17T^{2} \) |
| 19 | \( 1 + 2.53T + 19T^{2} \) |
| 23 | \( 1 + 5.50T + 23T^{2} \) |
| 29 | \( 1 - 6.49T + 29T^{2} \) |
| 31 | \( 1 + 5.78T + 31T^{2} \) |
| 37 | \( 1 + 6.24iT - 37T^{2} \) |
| 41 | \( 1 - 11.5iT - 41T^{2} \) |
| 43 | \( 1 + 7.55T + 43T^{2} \) |
| 47 | \( 1 - 0.158iT - 47T^{2} \) |
| 53 | \( 1 - 0.469iT - 53T^{2} \) |
| 59 | \( 1 - 6.99T + 59T^{2} \) |
| 61 | \( 1 + 3.90iT - 61T^{2} \) |
| 67 | \( 1 + 10.7T + 67T^{2} \) |
| 71 | \( 1 - 3.31iT - 71T^{2} \) |
| 73 | \( 1 + 0.691T + 73T^{2} \) |
| 79 | \( 1 - 10.1iT - 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 4.26iT - 89T^{2} \) |
| 97 | \( 1 + 2.82T + 97T^{2} \) |
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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
−8.623087230630682278857378380008, −7.87225479923650688140574286508, −7.52346012007628812831788036490, −6.64702731585397901152138462301, −5.34237670856327526171302143849, −4.74383256553682853937925713006, −3.94260465540199075139254026633, −2.50225215132972962051692459206, −1.45425227872908075004881852295, −0.03895517138488961371084070100,
2.11187974283616667447774121616, 3.00611326771041623739908871142, 4.09145891165338932873433328731, 4.85109327092215507299567407531, 5.65186454462631658344313821157, 6.76243141348932474680453293300, 7.60483252384289122714955033797, 8.216542635011996368455367120267, 8.994368488571984916291865263469, 10.12587039982624868714296167946
