| L(s) = 1 | + 2-s + 3-s + 4-s − 1.40·5-s + 6-s − 3.20·7-s + 8-s + 9-s − 1.40·10-s + 2.57·11-s + 12-s + 1.48·13-s − 3.20·14-s − 1.40·15-s + 16-s − 6.44·17-s + 18-s + 4.60·19-s − 1.40·20-s − 3.20·21-s + 2.57·22-s − 5.48·23-s + 24-s − 3.01·25-s + 1.48·26-s + 27-s − 3.20·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.630·5-s + 0.408·6-s − 1.21·7-s + 0.353·8-s + 0.333·9-s − 0.445·10-s + 0.776·11-s + 0.288·12-s + 0.412·13-s − 0.857·14-s − 0.363·15-s + 0.250·16-s − 1.56·17-s + 0.235·18-s + 1.05·19-s − 0.315·20-s − 0.700·21-s + 0.548·22-s − 1.14·23-s + 0.204·24-s − 0.602·25-s + 0.291·26-s + 0.192·27-s − 0.606·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5766 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5766 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 31 | \( 1 \) |
| good | 5 | \( 1 + 1.40T + 5T^{2} \) |
| 7 | \( 1 + 3.20T + 7T^{2} \) |
| 11 | \( 1 - 2.57T + 11T^{2} \) |
| 13 | \( 1 - 1.48T + 13T^{2} \) |
| 17 | \( 1 + 6.44T + 17T^{2} \) |
| 19 | \( 1 - 4.60T + 19T^{2} \) |
| 23 | \( 1 + 5.48T + 23T^{2} \) |
| 29 | \( 1 + 1.70T + 29T^{2} \) |
| 37 | \( 1 - 4.87T + 37T^{2} \) |
| 41 | \( 1 + 2.43T + 41T^{2} \) |
| 43 | \( 1 + 7.24T + 43T^{2} \) |
| 47 | \( 1 + 1.44T + 47T^{2} \) |
| 53 | \( 1 - 9.07T + 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 + 15.0T + 61T^{2} \) |
| 67 | \( 1 - 8.81T + 67T^{2} \) |
| 71 | \( 1 + 10.6T + 71T^{2} \) |
| 73 | \( 1 - 7.84T + 73T^{2} \) |
| 79 | \( 1 + 4.36T + 79T^{2} \) |
| 83 | \( 1 - 4.54T + 83T^{2} \) |
| 89 | \( 1 - 0.476T + 89T^{2} \) |
| 97 | \( 1 + 13.8T + 97T^{2} \) |
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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
−7.70001818682967969827823773477, −6.88635470710596816924286083242, −6.42350040081654236072137133470, −5.71625706710106842623623412908, −4.55840108421922906704418519687, −3.93203158099700418660418764150, −3.43059485566282133519623291254, −2.61591778719503048401052302742, −1.56768782092912382749688631088, 0,
1.56768782092912382749688631088, 2.61591778719503048401052302742, 3.43059485566282133519623291254, 3.93203158099700418660418764150, 4.55840108421922906704418519687, 5.71625706710106842623623412908, 6.42350040081654236072137133470, 6.88635470710596816924286083242, 7.70001818682967969827823773477
