L(s) = 1 | + 0.164·2-s + (1.34 − 1.08i)3-s − 1.97·4-s − i·5-s + (0.221 − 0.179i)6-s + i·7-s − 0.654·8-s + (0.630 − 2.93i)9-s − 0.164i·10-s + (3.08 − 1.22i)11-s + (−2.65 + 2.14i)12-s + 1.60i·13-s + 0.164i·14-s + (−1.08 − 1.34i)15-s + 3.83·16-s − 5.06·17-s + ⋯ |
L(s) = 1 | + 0.116·2-s + (0.777 − 0.628i)3-s − 0.986·4-s − 0.447i·5-s + (0.0906 − 0.0732i)6-s + 0.377i·7-s − 0.231·8-s + (0.210 − 0.977i)9-s − 0.0520i·10-s + (0.929 − 0.369i)11-s + (−0.767 + 0.619i)12-s + 0.445i·13-s + 0.0440i·14-s + (−0.281 − 0.347i)15-s + 0.959·16-s − 1.22·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.490 + 0.871i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.490 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.492524573\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.492524573\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.34 + 1.08i)T \) |
| 5 | \( 1 + iT \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 + (-3.08 + 1.22i)T \) |
good | 2 | \( 1 - 0.164T + 2T^{2} \) |
| 13 | \( 1 - 1.60iT - 13T^{2} \) |
| 17 | \( 1 + 5.06T + 17T^{2} \) |
| 19 | \( 1 + 5.56iT - 19T^{2} \) |
| 23 | \( 1 + 6.91iT - 23T^{2} \) |
| 29 | \( 1 - 9.80T + 29T^{2} \) |
| 31 | \( 1 + 5.21T + 31T^{2} \) |
| 37 | \( 1 + 11.6T + 37T^{2} \) |
| 41 | \( 1 + 3.68T + 41T^{2} \) |
| 43 | \( 1 + 0.866iT - 43T^{2} \) |
| 47 | \( 1 - 2.37iT - 47T^{2} \) |
| 53 | \( 1 + 6.59iT - 53T^{2} \) |
| 59 | \( 1 + 0.691iT - 59T^{2} \) |
| 61 | \( 1 + 9.81iT - 61T^{2} \) |
| 67 | \( 1 - 5.99T + 67T^{2} \) |
| 71 | \( 1 + 7.38iT - 71T^{2} \) |
| 73 | \( 1 - 16.3iT - 73T^{2} \) |
| 79 | \( 1 + 8.79iT - 79T^{2} \) |
| 83 | \( 1 + 2.46T + 83T^{2} \) |
| 89 | \( 1 - 7.69iT - 89T^{2} \) |
| 97 | \( 1 - 1.23T + 97T^{2} \) |
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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
−9.023718404768070503481571877506, −8.839272552846041122288141941100, −8.286142874124697792497845247230, −6.87813648309557504386426175020, −6.41027263061959417970095856150, −5.01203139369414061423097484244, −4.29508873853266767223720643732, −3.27654166533953763487191718574, −2.03634702133017058371918393790, −0.59624377808551325635302506332,
1.68925941468716607972861355343, 3.25531822337859046006368635990, 3.87619146208522426868171492014, 4.66758360386914419388725060324, 5.65061143228460469450119158302, 6.86566242344108164330572951950, 7.75995136139049950856575957828, 8.645118715421660061006755891144, 9.165638842790093151354836380401, 10.11467817540159038588734209740