L(s) = 1 | + (−0.5 + 0.866i)2-s + (0.866 − 0.5i)3-s + (−0.499 − 0.866i)4-s + (1.05 − 1.96i)5-s + 0.999i·6-s + (−3.29 + 1.90i)7-s + 0.999·8-s + (0.499 − 0.866i)9-s + (1.17 + 1.90i)10-s − 3.84·11-s + (−0.866 − 0.499i)12-s + (1.22 + 2.12i)13-s − 3.80i·14-s + (−0.0669 − 2.23i)15-s + (−0.5 + 0.866i)16-s + (−3.42 + 5.93i)17-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.499 − 0.288i)3-s + (−0.249 − 0.433i)4-s + (0.473 − 0.880i)5-s + 0.408i·6-s + (−1.24 + 0.718i)7-s + 0.353·8-s + (0.166 − 0.288i)9-s + (0.371 + 0.601i)10-s − 1.16·11-s + (−0.249 − 0.144i)12-s + (0.339 + 0.588i)13-s − 1.01i·14-s + (−0.0172 − 0.577i)15-s + (−0.125 + 0.216i)16-s + (−0.831 + 1.43i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.845 - 0.534i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.845 - 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5624175797\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5624175797\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 + (-0.866 + 0.5i)T \) |
| 5 | \( 1 + (-1.05 + 1.96i)T \) |
| 37 | \( 1 + (-4.13 - 4.46i)T \) |
good | 7 | \( 1 + (3.29 - 1.90i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + 3.84T + 11T^{2} \) |
| 13 | \( 1 + (-1.22 - 2.12i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (3.42 - 5.93i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.996 - 0.575i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 0.0656T + 23T^{2} \) |
| 29 | \( 1 + 2.70iT - 29T^{2} \) |
| 31 | \( 1 - 8.12iT - 31T^{2} \) |
| 41 | \( 1 + (-0.239 - 0.415i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + 10.0T + 43T^{2} \) |
| 47 | \( 1 - 1.94iT - 47T^{2} \) |
| 53 | \( 1 + (-4.83 - 2.78i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.87 - 1.08i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4.07 - 2.35i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6.70 - 3.87i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (3.72 + 6.44i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 6.82iT - 73T^{2} \) |
| 79 | \( 1 + (-2.57 + 1.48i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.83 - 2.79i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-6.89 - 3.98i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 2.90T + 97T^{2} \) |
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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
−9.944397790240271078508223763538, −9.099172414715419281432675166436, −8.605168562616892073337087733984, −7.962088307751293204773059658539, −6.65596785741149789393427047111, −6.17853847874233207237191365137, −5.27477988980504196212268821645, −4.15234118202279608476138204644, −2.79000079569503026998411701129, −1.65286100892657334150833526632,
0.24403103589138299602402203729, 2.35399744689749040260937396706, 2.98486226785687998535090503294, 3.80320028269145233993764667533, 5.07573205036142966850678256068, 6.28395016927738017904942400547, 7.17709052371651805936418205432, 7.81685037190452163675463222461, 9.007845725623462272726084822129, 9.686983099531940033756939876752