| L(s) = 1 | + 2.47·2-s + 0.222·3-s + 4.10·4-s + 3.30·5-s + 0.550·6-s − 0.323·7-s + 5.21·8-s − 2.95·9-s + 8.17·10-s − 0.193·11-s + 0.914·12-s − 13-s − 0.799·14-s + 0.736·15-s + 4.66·16-s + 6.34·17-s − 7.29·18-s − 3.55·19-s + 13.5·20-s − 0.0720·21-s − 0.477·22-s − 0.477·23-s + 1.16·24-s + 5.95·25-s − 2.47·26-s − 1.32·27-s − 1.32·28-s + ⋯ |
| L(s) = 1 | + 1.74·2-s + 0.128·3-s + 2.05·4-s + 1.47·5-s + 0.224·6-s − 0.122·7-s + 1.84·8-s − 0.983·9-s + 2.58·10-s − 0.0582·11-s + 0.264·12-s − 0.277·13-s − 0.213·14-s + 0.190·15-s + 1.16·16-s + 1.53·17-s − 1.71·18-s − 0.815·19-s + 3.04·20-s − 0.0157·21-s − 0.101·22-s − 0.0995·23-s + 0.236·24-s + 1.19·25-s − 0.484·26-s − 0.254·27-s − 0.251·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.715118601\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.715118601\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 - 2.47T + 2T^{2} \) |
| 3 | \( 1 - 0.222T + 3T^{2} \) |
| 5 | \( 1 - 3.30T + 5T^{2} \) |
| 7 | \( 1 + 0.323T + 7T^{2} \) |
| 11 | \( 1 + 0.193T + 11T^{2} \) |
| 17 | \( 1 - 6.34T + 17T^{2} \) |
| 19 | \( 1 + 3.55T + 19T^{2} \) |
| 23 | \( 1 + 0.477T + 23T^{2} \) |
| 29 | \( 1 - 8.11T + 29T^{2} \) |
| 31 | \( 1 + 0.307T + 31T^{2} \) |
| 37 | \( 1 + 5.06T + 37T^{2} \) |
| 41 | \( 1 + 5.23T + 41T^{2} \) |
| 43 | \( 1 - 6.63T + 43T^{2} \) |
| 47 | \( 1 + 7.94T + 47T^{2} \) |
| 53 | \( 1 + 5.71T + 53T^{2} \) |
| 59 | \( 1 + 7.42T + 59T^{2} \) |
| 61 | \( 1 - 3.27T + 61T^{2} \) |
| 67 | \( 1 + 9.73T + 67T^{2} \) |
| 71 | \( 1 + 8.95T + 71T^{2} \) |
| 73 | \( 1 - 14.8T + 73T^{2} \) |
| 79 | \( 1 - 13.7T + 79T^{2} \) |
| 83 | \( 1 + 10.6T + 83T^{2} \) |
| 89 | \( 1 - 2.15T + 89T^{2} \) |
| 97 | \( 1 + 4.66T + 97T^{2} \) |
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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
−9.855023102048797987986452778885, −8.835101917479644702147322304423, −7.80107255409043030897604424753, −6.55447812017849045376282196686, −6.12307771702584307260877780134, −5.37548263217310902379652264078, −4.76865353372778067085123388118, −3.38945667116114607319131018348, −2.74169544230741185542808887327, −1.75978716023601899573363403001,
1.75978716023601899573363403001, 2.74169544230741185542808887327, 3.38945667116114607319131018348, 4.76865353372778067085123388118, 5.37548263217310902379652264078, 6.12307771702584307260877780134, 6.55447812017849045376282196686, 7.80107255409043030897604424753, 8.835101917479644702147322304423, 9.855023102048797987986452778885
