L(s) = 1 | + (−2.36 + 2.36i)3-s + (2.19 + 0.429i)5-s + (2.93 − 2.93i)7-s − 8.20i·9-s − 1.20i·11-s + (2.14 − 2.14i)13-s + (−6.21 + 4.17i)15-s + (3.74 − 3.74i)17-s − 5.51·19-s + 13.8i·21-s + (4.77 + 0.475i)23-s + (4.63 + 1.88i)25-s + (12.3 + 12.3i)27-s − 1.52i·29-s − 5.73·31-s + ⋯ |
L(s) = 1 | + (−1.36 + 1.36i)3-s + (0.981 + 0.191i)5-s + (1.10 − 1.10i)7-s − 2.73i·9-s − 0.363i·11-s + (0.595 − 0.595i)13-s + (−1.60 + 1.07i)15-s + (0.909 − 0.909i)17-s − 1.26·19-s + 3.02i·21-s + (0.995 + 0.0990i)23-s + (0.926 + 0.376i)25-s + (2.37 + 2.37i)27-s − 0.283i·29-s − 1.02·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.18478 + 0.155593i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.18478 + 0.155593i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2.19 - 0.429i)T \) |
| 23 | \( 1 + (-4.77 - 0.475i)T \) |
good | 3 | \( 1 + (2.36 - 2.36i)T - 3iT^{2} \) |
| 7 | \( 1 + (-2.93 + 2.93i)T - 7iT^{2} \) |
| 11 | \( 1 + 1.20iT - 11T^{2} \) |
| 13 | \( 1 + (-2.14 + 2.14i)T - 13iT^{2} \) |
| 17 | \( 1 + (-3.74 + 3.74i)T - 17iT^{2} \) |
| 19 | \( 1 + 5.51T + 19T^{2} \) |
| 29 | \( 1 + 1.52iT - 29T^{2} \) |
| 31 | \( 1 + 5.73T + 31T^{2} \) |
| 37 | \( 1 + (3.78 - 3.78i)T - 37iT^{2} \) |
| 41 | \( 1 + 0.563T + 41T^{2} \) |
| 43 | \( 1 + (-5.82 - 5.82i)T + 43iT^{2} \) |
| 47 | \( 1 + (4.05 + 4.05i)T + 47iT^{2} \) |
| 53 | \( 1 + (-1.04 - 1.04i)T + 53iT^{2} \) |
| 59 | \( 1 + 0.826iT - 59T^{2} \) |
| 61 | \( 1 + 0.187iT - 61T^{2} \) |
| 67 | \( 1 + (6.63 - 6.63i)T - 67iT^{2} \) |
| 71 | \( 1 + 4.26T + 71T^{2} \) |
| 73 | \( 1 + (3.06 - 3.06i)T - 73iT^{2} \) |
| 79 | \( 1 - 11.9T + 79T^{2} \) |
| 83 | \( 1 + (-1.42 - 1.42i)T + 83iT^{2} \) |
| 89 | \( 1 - 3.10T + 89T^{2} \) |
| 97 | \( 1 + (3.19 - 3.19i)T - 97iT^{2} \) |
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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
−10.89297732999024037912951053893, −10.45407518092010819519587662195, −9.670820257858753555016488892883, −8.649285991178784425510732353131, −7.17047372786627599889492143320, −6.10523329386519919733762317932, −5.29452719789936351062817195481, −4.57834739137722234028024719354, −3.41867964623519037391612610795, −1.02851626257152879598535008600,
1.51882769249173467539216538421, 2.10209462744129983749547020835, 4.78098045092466976027750818602, 5.60087230763272354496333360745, 6.14640227015258143573652568341, 7.12611809229742744198016507010, 8.234852707141057484887102485678, 9.020733271280379520159788554425, 10.62403140199190807881364334678, 11.01642385456725208369396303685