| L(s) = 1 | + 5-s − 3.83·7-s + 1.70·11-s + 3.70·13-s − 1.70·17-s + 0.292·19-s + 5.83·23-s + 25-s − 8.67·29-s + 0.292·31-s − 3.83·35-s + 11.9·37-s + 6.96·41-s + 3.70·43-s − 1.87·47-s + 7.70·49-s + 11.6·53-s + 1.70·55-s − 11.6·59-s + 14.9·61-s + 3.70·65-s + 9.54·67-s − 15.9·71-s + 8·73-s − 6.54·77-s + 2·79-s + 11.8·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.44·7-s + 0.514·11-s + 1.02·13-s − 0.414·17-s + 0.0671·19-s + 1.21·23-s + 0.200·25-s − 1.60·29-s + 0.0525·31-s − 0.648·35-s + 1.96·37-s + 1.08·41-s + 0.565·43-s − 0.273·47-s + 1.10·49-s + 1.60·53-s + 0.230·55-s − 1.51·59-s + 1.91·61-s + 0.459·65-s + 1.16·67-s − 1.89·71-s + 0.936·73-s − 0.746·77-s + 0.225·79-s + 1.29·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.656334084\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.656334084\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 7 | \( 1 + 3.83T + 7T^{2} \) |
| 11 | \( 1 - 1.70T + 11T^{2} \) |
| 13 | \( 1 - 3.70T + 13T^{2} \) |
| 17 | \( 1 + 1.70T + 17T^{2} \) |
| 19 | \( 1 - 0.292T + 19T^{2} \) |
| 23 | \( 1 - 5.83T + 23T^{2} \) |
| 29 | \( 1 + 8.67T + 29T^{2} \) |
| 31 | \( 1 - 0.292T + 31T^{2} \) |
| 37 | \( 1 - 11.9T + 37T^{2} \) |
| 41 | \( 1 - 6.96T + 41T^{2} \) |
| 43 | \( 1 - 3.70T + 43T^{2} \) |
| 47 | \( 1 + 1.87T + 47T^{2} \) |
| 53 | \( 1 - 11.6T + 53T^{2} \) |
| 59 | \( 1 + 11.6T + 59T^{2} \) |
| 61 | \( 1 - 14.9T + 61T^{2} \) |
| 67 | \( 1 - 9.54T + 67T^{2} \) |
| 71 | \( 1 + 15.9T + 71T^{2} \) |
| 73 | \( 1 - 8T + 73T^{2} \) |
| 79 | \( 1 - 2T + 79T^{2} \) |
| 83 | \( 1 - 11.8T + 83T^{2} \) |
| 89 | \( 1 - 3T + 89T^{2} \) |
| 97 | \( 1 + 3.67T + 97T^{2} \) |
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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
−9.220440900413068302341972361121, −9.009224717526840745654640734219, −7.72650650548502505605044639274, −6.80191744740201368400744088527, −6.21322461261560821890840442186, −5.53129011503517137456249079935, −4.18964421794155493146012135937, −3.41646283974035447308382923512, −2.41358526328815336640970744784, −0.916572009131371766009966514180,
0.916572009131371766009966514180, 2.41358526328815336640970744784, 3.41646283974035447308382923512, 4.18964421794155493146012135937, 5.53129011503517137456249079935, 6.21322461261560821890840442186, 6.80191744740201368400744088527, 7.72650650548502505605044639274, 9.009224717526840745654640734219, 9.220440900413068302341972361121
