L(s) = 1 | + (0.5 + 0.866i)5-s + (1.5 − 2.59i)7-s + (2.5 − 4.33i)11-s + (−2 − 3.46i)13-s − 8·17-s − 2·19-s + (1 + 1.73i)23-s + (2 − 3.46i)25-s + (−3 + 5.19i)29-s + (−3.5 − 6.06i)31-s + 3·35-s − 6·37-s + (3 + 5.19i)41-s + (−1 + 1.73i)43-s + (3 − 5.19i)47-s + ⋯ |
L(s) = 1 | + (0.223 + 0.387i)5-s + (0.566 − 0.981i)7-s + (0.753 − 1.30i)11-s + (−0.554 − 0.960i)13-s − 1.94·17-s − 0.458·19-s + (0.208 + 0.361i)23-s + (0.400 − 0.692i)25-s + (−0.557 + 0.964i)29-s + (−0.628 − 1.08i)31-s + 0.507·35-s − 0.986·37-s + (0.468 + 0.811i)41-s + (−0.152 + 0.264i)43-s + (0.437 − 0.757i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.389247429\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.389247429\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.5 - 0.866i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-1.5 + 2.59i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.5 + 4.33i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2 + 3.46i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 8T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + (-1 - 1.73i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (3.5 + 6.06i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 + (-3 - 5.19i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1 - 1.73i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 5T + 53T^{2} \) |
| 59 | \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4 + 6.92i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (5 + 8.66i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - T + 73T^{2} \) |
| 79 | \( 1 + (-8 + 13.8i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (5.5 - 9.52i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + (-0.5 + 0.866i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.349616137934282788151949832386, −8.631773009216457655289104148310, −7.80505830461215580922561570875, −6.90338699782206882538814898869, −6.25634209951146386031091629133, −5.17144354159188181379688716291, −4.20232803554501897870850557385, −3.30706477490523308440198843026, −2.05622063501945365174930143288, −0.55381338827197526572741634589,
1.81070112000778147368059967004, 2.31488220422973967779953729488, 4.13802191171115778905610912905, 4.70485412316276148733355642903, 5.60418252433387890042564675962, 6.77784400855752444409487585816, 7.19065266565793287183054619160, 8.665264483932739708543942344117, 8.947918434215541808881819641105, 9.620884626245311715179437901058