L(s) = 1 | + (0.766 + 0.642i)2-s + (−1.65 − 0.513i)3-s + (0.173 + 0.984i)4-s + (0.939 + 0.342i)5-s + (−0.936 − 1.45i)6-s + (−0.500 + 2.83i)7-s + (−0.500 + 0.866i)8-s + (2.47 + 1.70i)9-s + (0.5 + 0.866i)10-s + (0.527 − 0.191i)11-s + (0.218 − 1.71i)12-s + (−1.19 + 1.00i)13-s + (−2.20 + 1.85i)14-s + (−1.37 − 1.04i)15-s + (−0.939 + 0.342i)16-s + (3.91 + 6.78i)17-s + ⋯ |
L(s) = 1 | + (0.541 + 0.454i)2-s + (−0.954 − 0.296i)3-s + (0.0868 + 0.492i)4-s + (0.420 + 0.152i)5-s + (−0.382 − 0.594i)6-s + (−0.189 + 1.07i)7-s + (−0.176 + 0.306i)8-s + (0.823 + 0.566i)9-s + (0.158 + 0.273i)10-s + (0.159 − 0.0578i)11-s + (0.0631 − 0.495i)12-s + (−0.331 + 0.277i)13-s + (−0.590 + 0.495i)14-s + (−0.355 − 0.270i)15-s + (−0.234 + 0.0855i)16-s + (0.950 + 1.64i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0891 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0891 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.945963 + 0.865047i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.945963 + 0.865047i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.766 - 0.642i)T \) |
| 3 | \( 1 + (1.65 + 0.513i)T \) |
| 5 | \( 1 + (-0.939 - 0.342i)T \) |
good | 7 | \( 1 + (0.500 - 2.83i)T + (-6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (-0.527 + 0.191i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (1.19 - 1.00i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-3.91 - 6.78i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.0937 + 0.162i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.230 - 1.30i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-0.737 - 0.618i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (1.56 + 8.89i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (4.25 + 7.37i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.59 + 4.69i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (2.71 - 0.988i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.892 + 5.06i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 + (9.19 + 3.34i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.18 + 6.74i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (4.70 - 3.94i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-8.25 - 14.2i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.61 + 2.79i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.20 - 4.36i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-3.56 - 2.98i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-1.46 + 2.53i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (17.5 - 6.38i)T + (74.3 - 62.3i)T^{2} \) |
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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
−12.36620883902665790687306669936, −11.44465810097910282694349748159, −10.40163147730718930345897966006, −9.297913289267575640717080899922, −8.050455349404028648090963654678, −6.92603737145014060764263846093, −5.85885176657214636248726207156, −5.50354877526879969630464590616, −3.94919464761537237557142674555, −2.11212786964324571667116316688,
1.01575698533363063759139900560, 3.21891424992223253642519712896, 4.57446710273340837861478428274, 5.33097898275059527612045186255, 6.56135899711482626885995839747, 7.42857268117862484119644867144, 9.296071516365400147483917639337, 10.13902486217940971396152855205, 10.69392206339350099563983834189, 11.82371262659428987213352208731