| L(s) = 1 | + 3-s − 2.60·5-s + 7-s + 9-s − 3.69·11-s − 5.98·13-s − 2.60·15-s − 5.23·17-s − 3.69·19-s + 21-s + 23-s + 1.77·25-s + 27-s − 4.36·29-s − 4.00·31-s − 3.69·33-s − 2.60·35-s − 8.22·37-s − 5.98·39-s + 6.55·41-s + 8.60·43-s − 2.60·45-s + 2.68·47-s + 49-s − 5.23·51-s + 11.3·53-s + 9.62·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.16·5-s + 0.377·7-s + 0.333·9-s − 1.11·11-s − 1.65·13-s − 0.672·15-s − 1.26·17-s − 0.847·19-s + 0.218·21-s + 0.208·23-s + 0.355·25-s + 0.192·27-s − 0.811·29-s − 0.718·31-s − 0.643·33-s − 0.440·35-s − 1.35·37-s − 0.958·39-s + 1.02·41-s + 1.31·43-s − 0.388·45-s + 0.391·47-s + 0.142·49-s − 0.732·51-s + 1.55·53-s + 1.29·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8732631016\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8732631016\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 2.60T + 5T^{2} \) |
| 11 | \( 1 + 3.69T + 11T^{2} \) |
| 13 | \( 1 + 5.98T + 13T^{2} \) |
| 17 | \( 1 + 5.23T + 17T^{2} \) |
| 19 | \( 1 + 3.69T + 19T^{2} \) |
| 29 | \( 1 + 4.36T + 29T^{2} \) |
| 31 | \( 1 + 4.00T + 31T^{2} \) |
| 37 | \( 1 + 8.22T + 37T^{2} \) |
| 41 | \( 1 - 6.55T + 41T^{2} \) |
| 43 | \( 1 - 8.60T + 43T^{2} \) |
| 47 | \( 1 - 2.68T + 47T^{2} \) |
| 53 | \( 1 - 11.3T + 53T^{2} \) |
| 59 | \( 1 - 14.6T + 59T^{2} \) |
| 61 | \( 1 - 4.08T + 61T^{2} \) |
| 67 | \( 1 + 3.79T + 67T^{2} \) |
| 71 | \( 1 - 1.77T + 71T^{2} \) |
| 73 | \( 1 - 1.52T + 73T^{2} \) |
| 79 | \( 1 + 1.97T + 79T^{2} \) |
| 83 | \( 1 - 7.08T + 83T^{2} \) |
| 89 | \( 1 - 12.9T + 89T^{2} \) |
| 97 | \( 1 - 1.66T + 97T^{2} \) |
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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
−7.68681053454511799232212459465, −7.41892764348497186980502218500, −6.82901933142069208200925294014, −5.59049201517829819362746573248, −4.91496711917427438318925141019, −4.24154107566307711939784291266, −3.65351657022409757064871374440, −2.42584800887570587652722268481, −2.22959391727995647894241412268, −0.41908913781131534970223846344,
0.41908913781131534970223846344, 2.22959391727995647894241412268, 2.42584800887570587652722268481, 3.65351657022409757064871374440, 4.24154107566307711939784291266, 4.91496711917427438318925141019, 5.59049201517829819362746573248, 6.82901933142069208200925294014, 7.41892764348497186980502218500, 7.68681053454511799232212459465
