| L(s) = 1 | + 5-s + 0.514·7-s + 3.32·11-s − 1.32·13-s − 3.32·17-s + 1.32·19-s − 4.12·23-s + 25-s − 1.38·29-s − 8.73·31-s + 0.514·35-s + 0.292·37-s − 11.3·41-s − 10.3·43-s + 4.86·47-s − 6.73·49-s − 5.02·53-s + 3.32·55-s + 5.02·59-s + 7.34·61-s − 1.32·65-s − 9.44·67-s − 8.99·71-s + 6.05·73-s + 1.70·77-s + 8.05·79-s − 1.54·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.194·7-s + 1.00·11-s − 0.366·13-s − 0.805·17-s + 0.303·19-s − 0.860·23-s + 0.200·25-s − 0.257·29-s − 1.56·31-s + 0.0869·35-s + 0.0481·37-s − 1.77·41-s − 1.57·43-s + 0.709·47-s − 0.962·49-s − 0.690·53-s + 0.447·55-s + 0.654·59-s + 0.940·61-s − 0.163·65-s − 1.15·67-s − 1.06·71-s + 0.708·73-s + 0.194·77-s + 0.906·79-s − 0.169·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.514T + 7T^{2} \) |
| 11 | \( 1 - 3.32T + 11T^{2} \) |
| 13 | \( 1 + 1.32T + 13T^{2} \) |
| 17 | \( 1 + 3.32T + 17T^{2} \) |
| 19 | \( 1 - 1.32T + 19T^{2} \) |
| 23 | \( 1 + 4.12T + 23T^{2} \) |
| 29 | \( 1 + 1.38T + 29T^{2} \) |
| 31 | \( 1 + 8.73T + 31T^{2} \) |
| 37 | \( 1 - 0.292T + 37T^{2} \) |
| 41 | \( 1 + 11.3T + 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 - 4.86T + 47T^{2} \) |
| 53 | \( 1 + 5.02T + 53T^{2} \) |
| 59 | \( 1 - 5.02T + 59T^{2} \) |
| 61 | \( 1 - 7.34T + 61T^{2} \) |
| 67 | \( 1 + 9.44T + 67T^{2} \) |
| 71 | \( 1 + 8.99T + 71T^{2} \) |
| 73 | \( 1 - 6.05T + 73T^{2} \) |
| 79 | \( 1 - 8.05T + 79T^{2} \) |
| 83 | \( 1 + 1.54T + 83T^{2} \) |
| 89 | \( 1 + 3T + 89T^{2} \) |
| 97 | \( 1 + 12.2T + 97T^{2} \) |
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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
−7.60766967306413998079362742041, −6.85945367866195320532742699818, −6.36551637653939071979592680707, −5.49813949181671930612025771738, −4.85049019855600148771862771948, −3.97628800370832098310534042979, −3.27375094941645161405477952772, −2.09673202264185550841939696409, −1.51791294628394743942830573327, 0,
1.51791294628394743942830573327, 2.09673202264185550841939696409, 3.27375094941645161405477952772, 3.97628800370832098310534042979, 4.85049019855600148771862771948, 5.49813949181671930612025771738, 6.36551637653939071979592680707, 6.85945367866195320532742699818, 7.60766967306413998079362742041
