L(s) = 1 | − i·3-s − 2.64·7-s − 9-s + 1.51i·11-s + 3.87i·13-s + 3.31·17-s − 7.08i·19-s + 2.64i·21-s − 4.82·23-s + i·27-s − 2.18i·29-s + 7.36·31-s + 1.51·33-s + 7.87i·37-s + 3.87·39-s + ⋯ |
L(s) = 1 | − 0.577i·3-s − 0.998·7-s − 0.333·9-s + 0.456i·11-s + 1.07i·13-s + 0.803·17-s − 1.62i·19-s + 0.576i·21-s − 1.00·23-s + 0.192i·27-s − 0.405i·29-s + 1.32·31-s + 0.263·33-s + 1.29i·37-s + 0.619·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.318i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.451136222\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.451136222\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 2.64T + 7T^{2} \) |
| 11 | \( 1 - 1.51iT - 11T^{2} \) |
| 13 | \( 1 - 3.87iT - 13T^{2} \) |
| 17 | \( 1 - 3.31T + 17T^{2} \) |
| 19 | \( 1 + 7.08iT - 19T^{2} \) |
| 23 | \( 1 + 4.82T + 23T^{2} \) |
| 29 | \( 1 + 2.18iT - 29T^{2} \) |
| 31 | \( 1 - 7.36T + 31T^{2} \) |
| 37 | \( 1 - 7.87iT - 37T^{2} \) |
| 41 | \( 1 - 8.72T + 41T^{2} \) |
| 43 | \( 1 + 1.01iT - 43T^{2} \) |
| 47 | \( 1 - 7.08T + 47T^{2} \) |
| 53 | \( 1 - 4.50iT - 53T^{2} \) |
| 59 | \( 1 - 6.79iT - 59T^{2} \) |
| 61 | \( 1 + 3.60iT - 61T^{2} \) |
| 67 | \( 1 + 1.01iT - 67T^{2} \) |
| 71 | \( 1 - 6.72T + 71T^{2} \) |
| 73 | \( 1 - 15.5T + 73T^{2} \) |
| 79 | \( 1 - 7.36T + 79T^{2} \) |
| 83 | \( 1 + 7.74iT - 83T^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 - 11.1T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.045157984358803352914362327837, −8.055143406604106310796091804194, −7.31192131440921268728700779226, −6.54735641707656188606800181013, −6.12636039955725407700081250723, −4.92944873048515401704111303829, −4.09126826011184126008985000217, −2.97894481934235703887237635005, −2.17599943435635160806574549861, −0.77879348686956699590168710429,
0.74398117371210092262737769271, 2.43128340145661761596844591996, 3.46838988394947283658350777304, 3.86966211653493866683065667606, 5.20617190738389387383026705024, 5.88314782351001109771632358779, 6.42673546360306079418320552004, 7.76942306960994776842173358833, 8.086511302344515980232351432067, 9.161021473544472458876143679823