| L(s) = 1 | + 0.734·2-s − 1.46·4-s − 2.54·5-s − 2.54·8-s − 1.86·10-s − 3.71·11-s − 13-s + 1.05·16-s + 2.10·17-s + 5.92·19-s + 3.71·20-s − 2.72·22-s + 6.25·23-s + 1.46·25-s − 0.734·26-s + 8.83·29-s − 5.70·31-s + 5.85·32-s + 1.54·34-s + 6.56·37-s + 4.34·38-s + 6.46·40-s + 10.3·41-s − 10.6·43-s + 5.42·44-s + 4.59·46-s − 1.46·47-s + ⋯ |
| L(s) = 1 | + 0.519·2-s − 0.730·4-s − 1.13·5-s − 0.898·8-s − 0.590·10-s − 1.11·11-s − 0.277·13-s + 0.263·16-s + 0.510·17-s + 1.35·19-s + 0.830·20-s − 0.581·22-s + 1.30·23-s + 0.292·25-s − 0.144·26-s + 1.64·29-s − 1.02·31-s + 1.03·32-s + 0.265·34-s + 1.07·37-s + 0.705·38-s + 1.02·40-s + 1.60·41-s − 1.62·43-s + 0.817·44-s + 0.677·46-s − 0.214·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 0.734T + 2T^{2} \) |
| 5 | \( 1 + 2.54T + 5T^{2} \) |
| 11 | \( 1 + 3.71T + 11T^{2} \) |
| 17 | \( 1 - 2.10T + 17T^{2} \) |
| 19 | \( 1 - 5.92T + 19T^{2} \) |
| 23 | \( 1 - 6.25T + 23T^{2} \) |
| 29 | \( 1 - 8.83T + 29T^{2} \) |
| 31 | \( 1 + 5.70T + 31T^{2} \) |
| 37 | \( 1 - 6.56T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 + 10.6T + 43T^{2} \) |
| 47 | \( 1 + 1.46T + 47T^{2} \) |
| 53 | \( 1 + 6.45T + 53T^{2} \) |
| 59 | \( 1 - 11.4T + 59T^{2} \) |
| 61 | \( 1 + 9.11T + 61T^{2} \) |
| 67 | \( 1 + 14.0T + 67T^{2} \) |
| 71 | \( 1 + 14.4T + 71T^{2} \) |
| 73 | \( 1 + 6.14T + 73T^{2} \) |
| 79 | \( 1 + 7.96T + 79T^{2} \) |
| 83 | \( 1 + 12.1T + 83T^{2} \) |
| 89 | \( 1 - 11.8T + 89T^{2} \) |
| 97 | \( 1 - 7.10T + 97T^{2} \) |
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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
−7.62114191776986944147531144732, −7.37512163811093269320140406837, −6.16361764564285496282960959339, −5.36708187103592129714821518016, −4.80998709496067160529833003770, −4.20810644032088231903777862239, −3.14030918627271827810789315571, −2.93068857508690434543578864885, −1.07266925099821890727208850518, 0,
1.07266925099821890727208850518, 2.93068857508690434543578864885, 3.14030918627271827810789315571, 4.20810644032088231903777862239, 4.80998709496067160529833003770, 5.36708187103592129714821518016, 6.16361764564285496282960959339, 7.37512163811093269320140406837, 7.62114191776986944147531144732
