| L(s) = 1 | − 0.585·5-s + 0.585·7-s − 0.414·11-s − 2.24·13-s − 2.41·17-s + 2.58·19-s + 1.41·23-s − 4.65·25-s − 8.07·29-s + 3·31-s − 0.343·35-s + 7.48·37-s + 41-s + 5·43-s + 7.58·47-s − 6.65·49-s − 1.17·53-s + 0.242·55-s + 8.48·59-s + 6.65·61-s + 1.31·65-s + 6.48·67-s − 4.07·71-s + 12.3·73-s − 0.242·77-s − 3.65·79-s − 13.0·83-s + ⋯ |
| L(s) = 1 | − 0.261·5-s + 0.221·7-s − 0.124·11-s − 0.621·13-s − 0.585·17-s + 0.593·19-s + 0.294·23-s − 0.931·25-s − 1.49·29-s + 0.538·31-s − 0.0580·35-s + 1.23·37-s + 0.156·41-s + 0.762·43-s + 1.10·47-s − 0.950·49-s − 0.160·53-s + 0.0327·55-s + 1.10·59-s + 0.852·61-s + 0.162·65-s + 0.792·67-s − 0.483·71-s + 1.44·73-s − 0.0276·77-s − 0.411·79-s − 1.43·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5904 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.586201716\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.586201716\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 5 | \( 1 + 0.585T + 5T^{2} \) |
| 7 | \( 1 - 0.585T + 7T^{2} \) |
| 11 | \( 1 + 0.414T + 11T^{2} \) |
| 13 | \( 1 + 2.24T + 13T^{2} \) |
| 17 | \( 1 + 2.41T + 17T^{2} \) |
| 19 | \( 1 - 2.58T + 19T^{2} \) |
| 23 | \( 1 - 1.41T + 23T^{2} \) |
| 29 | \( 1 + 8.07T + 29T^{2} \) |
| 31 | \( 1 - 3T + 31T^{2} \) |
| 37 | \( 1 - 7.48T + 37T^{2} \) |
| 43 | \( 1 - 5T + 43T^{2} \) |
| 47 | \( 1 - 7.58T + 47T^{2} \) |
| 53 | \( 1 + 1.17T + 53T^{2} \) |
| 59 | \( 1 - 8.48T + 59T^{2} \) |
| 61 | \( 1 - 6.65T + 61T^{2} \) |
| 67 | \( 1 - 6.48T + 67T^{2} \) |
| 71 | \( 1 + 4.07T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 + 3.65T + 79T^{2} \) |
| 83 | \( 1 + 13.0T + 83T^{2} \) |
| 89 | \( 1 - 0.343T + 89T^{2} \) |
| 97 | \( 1 - 16.2T + 97T^{2} \) |
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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
−7.928732087449375836877058955612, −7.50533679347545568779084225444, −6.77763428815163462176889720803, −5.87594442622314570742042707219, −5.25212658424128580058495675789, −4.40069095185935136763848462130, −3.75435578352255912087970176559, −2.71812200311317019952825442784, −1.94663762301278185322338184551, −0.65240007600268458310014122672,
0.65240007600268458310014122672, 1.94663762301278185322338184551, 2.71812200311317019952825442784, 3.75435578352255912087970176559, 4.40069095185935136763848462130, 5.25212658424128580058495675789, 5.87594442622314570742042707219, 6.77763428815163462176889720803, 7.50533679347545568779084225444, 7.928732087449375836877058955612
