| L(s) = 1 | + (0.900 − 0.433i)2-s + (1.16 + 1.27i)3-s + (0.623 − 0.781i)4-s + (1.29 + 1.19i)5-s + (1.60 + 0.646i)6-s + (0.993 − 2.45i)7-s + (0.222 − 0.974i)8-s + (−0.274 + 2.98i)9-s + (1.68 + 0.519i)10-s + (1.94 + 1.32i)11-s + (1.72 − 0.114i)12-s + (−1.57 − 1.07i)13-s + (−0.169 − 2.64i)14-s + (−0.0257 + 3.05i)15-s + (−0.222 − 0.974i)16-s + (1.42 − 0.214i)17-s + ⋯ |
| L(s) = 1 | + (0.637 − 0.306i)2-s + (0.673 + 0.738i)3-s + (0.311 − 0.390i)4-s + (0.577 + 0.536i)5-s + (0.656 + 0.263i)6-s + (0.375 − 0.926i)7-s + (0.0786 − 0.344i)8-s + (−0.0915 + 0.995i)9-s + (0.532 + 0.164i)10-s + (0.587 + 0.400i)11-s + (0.498 − 0.0331i)12-s + (−0.437 − 0.298i)13-s + (−0.0452 − 0.705i)14-s + (−0.00665 + 0.788i)15-s + (−0.0556 − 0.243i)16-s + (0.344 − 0.0519i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 - 0.243i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.969 - 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.21641 + 0.398365i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.21641 + 0.398365i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.900 + 0.433i)T \) |
| 3 | \( 1 + (-1.16 - 1.27i)T \) |
| 7 | \( 1 + (-0.993 + 2.45i)T \) |
| good | 5 | \( 1 + (-1.29 - 1.19i)T + (0.373 + 4.98i)T^{2} \) |
| 11 | \( 1 + (-1.94 - 1.32i)T + (4.01 + 10.2i)T^{2} \) |
| 13 | \( 1 + (1.57 + 1.07i)T + (4.74 + 12.1i)T^{2} \) |
| 17 | \( 1 + (-1.42 + 0.214i)T + (16.2 - 5.01i)T^{2} \) |
| 19 | \( 1 + (-0.450 - 0.780i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.544 - 1.38i)T + (-16.8 + 15.6i)T^{2} \) |
| 29 | \( 1 + (-1.27 + 0.191i)T + (27.7 - 8.54i)T^{2} \) |
| 31 | \( 1 + 2.74T + 31T^{2} \) |
| 37 | \( 1 + (0.761 - 1.93i)T + (-27.1 - 25.1i)T^{2} \) |
| 41 | \( 1 + (3.79 - 1.17i)T + (33.8 - 23.0i)T^{2} \) |
| 43 | \( 1 + (2.38 + 0.735i)T + (35.5 + 24.2i)T^{2} \) |
| 47 | \( 1 + (-0.189 + 0.0913i)T + (29.3 - 36.7i)T^{2} \) |
| 53 | \( 1 + (0.112 + 0.287i)T + (-38.8 + 36.0i)T^{2} \) |
| 59 | \( 1 + (-0.292 - 1.28i)T + (-53.1 + 25.5i)T^{2} \) |
| 61 | \( 1 + (-3.79 - 4.75i)T + (-13.5 + 59.4i)T^{2} \) |
| 67 | \( 1 - 4.08T + 67T^{2} \) |
| 71 | \( 1 + (-7.48 + 9.39i)T + (-15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (9.06 - 6.17i)T + (26.6 - 67.9i)T^{2} \) |
| 79 | \( 1 + 6.54T + 79T^{2} \) |
| 83 | \( 1 + (10.6 - 7.23i)T + (30.3 - 77.2i)T^{2} \) |
| 89 | \( 1 + (0.529 + 7.06i)T + (-88.0 + 13.2i)T^{2} \) |
| 97 | \( 1 + (-2.05 + 3.56i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.03974712985487587059814759460, −9.806642518331656993122862808723, −8.561443859381560981604189296986, −7.53419539367214966666964052503, −6.76260436554318598089615976273, −5.54945261542180632245504229008, −4.63220574517537097378529372289, −3.80456597535706543203716625505, −2.85803115326686473054879945624, −1.68958216994796049635536545641,
1.51890246692473643241978007452, 2.51729477908537194007783643494, 3.63969214020407972484635833388, 4.95395429823438499304596247839, 5.76410140002983531504837543313, 6.58715906970633010187448296585, 7.50057279612628510273622075117, 8.491911904732471944775157450392, 8.985660342731432660023043603970, 9.808982361791908529188428959791
