| L(s) = 1 | + (−2.42 + 0.881i)2-s + (0.345 − 1.69i)3-s + (3.55 − 2.98i)4-s + (0.173 + 0.984i)5-s + (0.660 + 4.41i)6-s + (−2.32 − 1.95i)7-s + (−3.40 + 5.89i)8-s + (−2.76 − 1.17i)9-s + (−1.28 − 2.23i)10-s + (0.601 − 3.40i)11-s + (−3.83 − 7.06i)12-s + (1.17 + 0.428i)13-s + (7.35 + 2.67i)14-s + (1.73 + 0.0450i)15-s + (1.43 − 8.12i)16-s + (−3.31 − 5.74i)17-s + ⋯ |
| L(s) = 1 | + (−1.71 + 0.623i)2-s + (0.199 − 0.979i)3-s + (1.77 − 1.49i)4-s + (0.0776 + 0.440i)5-s + (0.269 + 1.80i)6-s + (−0.880 − 0.738i)7-s + (−1.20 + 2.08i)8-s + (−0.920 − 0.390i)9-s + (−0.407 − 0.705i)10-s + (0.181 − 1.02i)11-s + (−1.10 − 2.03i)12-s + (0.326 + 0.118i)13-s + (1.96 + 0.715i)14-s + (0.447 + 0.0116i)15-s + (0.358 − 2.03i)16-s + (−0.804 − 1.39i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0382 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0382 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.304656 - 0.293220i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.304656 - 0.293220i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.345 + 1.69i)T \) |
| 5 | \( 1 + (-0.173 - 0.984i)T \) |
| good | 2 | \( 1 + (2.42 - 0.881i)T + (1.53 - 1.28i)T^{2} \) |
| 7 | \( 1 + (2.32 + 1.95i)T + (1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (-0.601 + 3.40i)T + (-10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (-1.17 - 0.428i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (3.31 + 5.74i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.19 + 2.07i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.52 + 2.95i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (6.64 - 2.42i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-1.15 + 0.972i)T + (5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (-4.55 - 7.89i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-6.73 - 2.44i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.259 + 1.47i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-7.70 - 6.46i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 - 8.41T + 53T^{2} \) |
| 59 | \( 1 + (-0.497 - 2.81i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (7.66 + 6.42i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (0.507 + 0.184i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (1.67 + 2.90i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (3.82 - 6.63i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.0342 + 0.0124i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-10.4 + 3.80i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (0.126 - 0.219i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.142 - 0.810i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−13.20901794750468240339347112365, −11.47105977827640113948260144681, −10.83455808593935558986900003184, −9.474635944577356032978844007497, −8.795170115558402951964394302301, −7.52243430114230265682125894329, −6.85809742096239471489783458901, −6.06328588633506465250653516409, −2.83043436968165674128840461518, −0.71762009684969935964632334829,
2.24907762726463509665833463274, 3.83277891597028478924919971151, 5.92470507678128462108100611815, 7.56809081668540424524498185359, 8.866000938075675362245588598362, 9.248467857348470771721262472080, 10.14290965322847378271106898954, 11.02078928352230904934596779859, 12.13287348465148244705512711481, 13.04168569248206476941165017386
