L(s) = 1 | − 3-s − 5-s + (1.92 − 3.34i)7-s + 9-s + (−1.93 + 3.34i)11-s + (1.12 + 1.94i)13-s + 15-s + (−2.67 − 4.63i)17-s + (0.0704 + 0.121i)19-s + (−1.92 + 3.34i)21-s + (1.89 + 3.28i)23-s + 25-s − 27-s + (0.794 − 1.37i)29-s + (3.48 − 6.03i)31-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + (0.728 − 1.26i)7-s + 0.333·9-s + (−0.582 + 1.00i)11-s + (0.311 + 0.540i)13-s + 0.258·15-s + (−0.649 − 1.12i)17-s + (0.0161 + 0.0279i)19-s + (−0.420 + 0.728i)21-s + (0.395 + 0.685i)23-s + 0.200·25-s − 0.192·27-s + (0.147 − 0.255i)29-s + (0.625 − 1.08i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.424 + 0.905i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.424 + 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.274069749\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.274069749\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 + (6.19 + 5.35i)T \) |
good | 7 | \( 1 + (-1.92 + 3.34i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.93 - 3.34i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.12 - 1.94i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (2.67 + 4.63i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.0704 - 0.121i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.89 - 3.28i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.794 + 1.37i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.48 + 6.03i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.60 - 2.78i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (2.14 - 3.71i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 - 9.83T + 43T^{2} \) |
| 47 | \( 1 + (2.54 - 4.40i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 12.6T + 53T^{2} \) |
| 59 | \( 1 + 9.44T + 59T^{2} \) |
| 61 | \( 1 + (0.809 + 1.40i)T + (-30.5 + 52.8i)T^{2} \) |
| 71 | \( 1 + (-4.60 + 7.96i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (0.483 + 0.838i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.61 + 2.80i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.64 - 4.58i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 12.9T + 89T^{2} \) |
| 97 | \( 1 + (-3.53 - 6.11i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.006958190859854893905208739042, −7.50905313752910913896419126307, −7.03214513423856335624864248638, −6.20023477145040872900030660930, −5.07575995781317003415197045571, −4.51316190275084825391214707788, −4.05693612508390785359483607568, −2.75067240258826304237618397599, −1.58784580423029196154677194510, −0.51828332756482777311249660972,
0.906150536688501834557171109363, 2.18573071255124778080281462689, 3.05684468037302717978480367636, 4.10250200984687929620431344468, 4.98343452762184836973257718000, 5.64861740740033617043024571766, 6.14100725528026235171246614206, 7.09219685386263766628384004975, 8.034119826618967194732115807395, 8.641311518994415049599818460799