| 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 | \( 1 + (0.142 + 0.810i)T + (-91.1 + 33.1i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.04168569248206476941165017386, −12.13287348465148244705512711481, −11.02078928352230904934596779859, −10.14290965322847378271106898954, −9.248467857348470771721262472080, −8.866000938075675362245588598362, −7.56809081668540424524498185359, −5.92470507678128462108100611815, −3.83277891597028478924919971151, −2.24907762726463509665833463274,
0.71762009684969935964632334829, 2.83043436968165674128840461518, 6.06328588633506465250653516409, 6.85809742096239471489783458901, 7.52243430114230265682125894329, 8.795170115558402951964394302301, 9.474635944577356032978844007497, 10.83455808593935558986900003184, 11.47105977827640113948260144681, 13.20901794750468240339347112365
