L(s) = 1 | + (0.866 − 0.5i)3-s + (−3.08 − 1.78i)5-s + (2.38 − 1.14i)7-s + (0.499 − 0.866i)9-s + (−3.52 + 2.03i)11-s − 1.44i·13-s − 3.56·15-s + (−3.49 − 6.05i)17-s + (−0.261 − 0.150i)19-s + (1.48 − 2.18i)21-s + (−1.21 + 2.10i)23-s + (3.85 + 6.67i)25-s − 0.999i·27-s + 0.151i·29-s + (−2.37 − 4.11i)31-s + ⋯ |
L(s) = 1 | + (0.499 − 0.288i)3-s + (−1.38 − 0.797i)5-s + (0.900 − 0.434i)7-s + (0.166 − 0.288i)9-s + (−1.06 + 0.613i)11-s − 0.399i·13-s − 0.920·15-s + (−0.847 − 1.46i)17-s + (−0.0599 − 0.0345i)19-s + (0.325 − 0.477i)21-s + (−0.252 + 0.437i)23-s + (0.771 + 1.33i)25-s − 0.192i·27-s + 0.0281i·29-s + (−0.426 − 0.739i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.777 + 0.629i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.777 + 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.315033 - 0.889483i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.315033 - 0.889483i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.866 + 0.5i)T \) |
| 7 | \( 1 + (-2.38 + 1.14i)T \) |
good | 5 | \( 1 + (3.08 + 1.78i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (3.52 - 2.03i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 1.44iT - 13T^{2} \) |
| 17 | \( 1 + (3.49 + 6.05i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.261 + 0.150i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.21 - 2.10i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 0.151iT - 29T^{2} \) |
| 31 | \( 1 + (2.37 + 4.11i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (9.82 + 5.67i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 0.239T + 41T^{2} \) |
| 43 | \( 1 + 1.32iT - 43T^{2} \) |
| 47 | \( 1 + (3.17 - 5.50i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.18 + 2.99i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-9.73 + 5.62i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.64 - 2.10i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.79 - 2.19i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 8.46T + 71T^{2} \) |
| 73 | \( 1 + (0.284 + 0.493i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.746 - 1.29i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 10.0iT - 83T^{2} \) |
| 89 | \( 1 + (-1.83 + 3.17i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−10.19047587681059956355666203085, −9.033294095762264777386680809355, −8.308938792303212637813009980529, −7.54771846403428006127884206980, −7.18275674042666581348781473919, −5.25693700764679160621374541204, −4.61839103977434257633926103600, −3.62549118059853357504140447498, −2.17250830642384288912351827226, −0.45777098782906771678383734954,
2.14832563836418715766775399619, 3.33957397085469045766657088167, 4.19394684408729226265389830083, 5.23539825409879439166708716463, 6.58412668564405387082825288326, 7.54770937786511690862263032717, 8.447085487309837099537546428202, 8.585980001263242175041385700961, 10.32390530084948219920049958217, 10.79174468344455418711962531450