L(s) = 1 | + 2.66·5-s + (0.654 + 2.56i)7-s − 3.98·11-s + (1.00 + 1.73i)13-s + (3.57 + 6.18i)17-s + (4.01 − 6.96i)19-s − 0.887·23-s + 2.12·25-s + (1.35 − 2.33i)29-s + (−0.614 + 1.06i)31-s + (1.74 + 6.84i)35-s + (5.26 − 9.11i)37-s + (1.43 + 2.48i)41-s + (−3.40 + 5.88i)43-s + (6.06 + 10.5i)47-s + ⋯ |
L(s) = 1 | + 1.19·5-s + (0.247 + 0.968i)7-s − 1.20·11-s + (0.277 + 0.480i)13-s + (0.866 + 1.50i)17-s + (0.922 − 1.59i)19-s − 0.185·23-s + 0.424·25-s + (0.250 − 0.434i)29-s + (−0.110 + 0.191i)31-s + (0.295 + 1.15i)35-s + (0.865 − 1.49i)37-s + (0.224 + 0.388i)41-s + (−0.518 + 0.898i)43-s + (0.885 + 1.53i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.485 - 0.874i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.485 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.294858755\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.294858755\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.654 - 2.56i)T \) |
good | 5 | \( 1 - 2.66T + 5T^{2} \) |
| 11 | \( 1 + 3.98T + 11T^{2} \) |
| 13 | \( 1 + (-1.00 - 1.73i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-3.57 - 6.18i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.01 + 6.96i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 0.887T + 23T^{2} \) |
| 29 | \( 1 + (-1.35 + 2.33i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (0.614 - 1.06i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.26 + 9.11i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.43 - 2.48i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.40 - 5.88i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-6.06 - 10.5i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.38 - 4.13i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (4.79 - 8.29i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.74 - 8.22i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.49 + 9.51i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 4.62T + 71T^{2} \) |
| 73 | \( 1 + (-2.01 - 3.48i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.514 + 0.891i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (5.26 - 9.12i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (1.72 - 2.99i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1.12 - 1.94i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.014916442082494583003869693186, −8.111462592002765482956749486015, −7.47040143463661316449259277446, −6.29800057864124037548656554884, −5.77846463227899206701582155057, −5.25330296206338472397851336844, −4.28071389447907846594020134129, −2.88527111635827832902383182469, −2.35093344777915352647645271639, −1.28771458337066008079552521844,
0.75715023264862408546192434942, 1.82966687220334302833978975023, 2.91746238692596341223345664215, 3.71189382469522460251802162463, 5.09919075970120280789633590277, 5.34357757651961231433704136825, 6.22690788906793880607098315605, 7.24526866144374029288307045987, 7.76249042879901650147671386800, 8.482403916539265000931988980811