L(s) = 1 | + (−0.5 − 1.65i)3-s + (2.18 + 1.26i)5-s − 3.37·7-s + (−2.5 + 1.65i)9-s + 3.46i·11-s + (2.87 − 1.65i)13-s + (1 − 4.25i)15-s + (2.18 + 1.26i)17-s + (4 + 1.73i)19-s + (1.68 + 5.59i)21-s + (7.93 − 4.57i)23-s + (0.686 + 1.18i)25-s + (4 + 3.31i)27-s + (−0.186 − 0.322i)29-s + 7.72i·31-s + ⋯ |
L(s) = 1 | + (−0.288 − 0.957i)3-s + (0.977 + 0.564i)5-s − 1.27·7-s + (−0.833 + 0.552i)9-s + 1.04i·11-s + (0.796 − 0.459i)13-s + (0.258 − 1.09i)15-s + (0.530 + 0.306i)17-s + (0.917 + 0.397i)19-s + (0.367 + 1.22i)21-s + (1.65 − 0.954i)23-s + (0.137 + 0.237i)25-s + (0.769 + 0.638i)27-s + (−0.0345 − 0.0598i)29-s + 1.38i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0179i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0179i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.50772 - 0.0134966i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.50772 - 0.0134966i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.5 + 1.65i)T \) |
| 19 | \( 1 + (-4 - 1.73i)T \) |
good | 5 | \( 1 + (-2.18 - 1.26i)T + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + 3.37T + 7T^{2} \) |
| 11 | \( 1 - 3.46iT - 11T^{2} \) |
| 13 | \( 1 + (-2.87 + 1.65i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.18 - 1.26i)T + (8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-7.93 + 4.57i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.186 + 0.322i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 7.72iT - 31T^{2} \) |
| 37 | \( 1 - 11.1iT - 37T^{2} \) |
| 41 | \( 1 + (-3.18 + 5.51i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.87 + 10.1i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.813 - 0.469i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2.81 + 4.87i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (0.813 - 1.40i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.24 + 0.718i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (6.18 - 10.7i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (2.87 - 4.97i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-12.9 - 7.49i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 3.46iT - 83T^{2} \) |
| 89 | \( 1 + (0.813 + 1.40i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.44 - 1.40i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.22956598199647540226676154327, −9.344091836795778145203780246234, −8.399752719197876835621716635077, −7.17285942522205425486296098798, −6.71938448756814750981145849552, −5.96815269680143070629186166598, −5.14559467704018874175874733711, −3.36061022765361375452663154638, −2.54393436579095325077830446798, −1.20312932012362480941499771938,
0.905688510322046202803204416781, 2.92498393573560519748166763561, 3.63345108687961436578393319518, 4.93302925441963677960241402278, 5.88011070414786089192942119330, 6.15755990880469641101682424949, 7.55248517864707559301896974335, 9.040210573114173729953037844010, 9.270871991507529608534615295414, 9.797732075067669589229870056131