| L(s) = 1 | + 1.85·2-s − 1.09·3-s + 1.44·4-s − 0.392·5-s − 2.03·6-s + 2.97·7-s − 1.03·8-s − 1.79·9-s − 0.728·10-s − 11-s − 1.58·12-s − 5.33·13-s + 5.52·14-s + 0.430·15-s − 4.80·16-s − 4.90·17-s − 3.33·18-s − 0.516·19-s − 0.567·20-s − 3.26·21-s − 1.85·22-s + 4.69·23-s + 1.13·24-s − 4.84·25-s − 9.89·26-s + 5.26·27-s + 4.29·28-s + ⋯ |
| L(s) = 1 | + 1.31·2-s − 0.633·3-s + 0.722·4-s − 0.175·5-s − 0.830·6-s + 1.12·7-s − 0.364·8-s − 0.599·9-s − 0.230·10-s − 0.301·11-s − 0.457·12-s − 1.47·13-s + 1.47·14-s + 0.111·15-s − 1.20·16-s − 1.18·17-s − 0.786·18-s − 0.118·19-s − 0.126·20-s − 0.712·21-s − 0.395·22-s + 0.979·23-s + 0.230·24-s − 0.969·25-s − 1.94·26-s + 1.01·27-s + 0.812·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 + T \) |
| 131 | \( 1 + T \) |
| good | 2 | \( 1 - 1.85T + 2T^{2} \) |
| 3 | \( 1 + 1.09T + 3T^{2} \) |
| 5 | \( 1 + 0.392T + 5T^{2} \) |
| 7 | \( 1 - 2.97T + 7T^{2} \) |
| 13 | \( 1 + 5.33T + 13T^{2} \) |
| 17 | \( 1 + 4.90T + 17T^{2} \) |
| 19 | \( 1 + 0.516T + 19T^{2} \) |
| 23 | \( 1 - 4.69T + 23T^{2} \) |
| 29 | \( 1 - 1.88T + 29T^{2} \) |
| 31 | \( 1 - 3.34T + 31T^{2} \) |
| 37 | \( 1 + 9.44T + 37T^{2} \) |
| 41 | \( 1 - 2.16T + 41T^{2} \) |
| 43 | \( 1 + 3.87T + 43T^{2} \) |
| 47 | \( 1 + 8.29T + 47T^{2} \) |
| 53 | \( 1 - 5.53T + 53T^{2} \) |
| 59 | \( 1 - 9.86T + 59T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 + 0.784T + 67T^{2} \) |
| 71 | \( 1 + 13.9T + 71T^{2} \) |
| 73 | \( 1 - 5.40T + 73T^{2} \) |
| 79 | \( 1 + 4.35T + 79T^{2} \) |
| 83 | \( 1 - 11.9T + 83T^{2} \) |
| 89 | \( 1 + 9.78T + 89T^{2} \) |
| 97 | \( 1 + 9.38T + 97T^{2} \) |
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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
−9.010510832697850693854317813090, −8.286555927848319300094549099989, −7.22837361877626827819253873605, −6.45406179414174046853970226782, −5.43682005869899241867952100413, −4.92506055608860632222633957732, −4.39735445788776990160902957860, −3.06379025583405731648168277982, −2.11906184612582144327160926013, 0,
2.11906184612582144327160926013, 3.06379025583405731648168277982, 4.39735445788776990160902957860, 4.92506055608860632222633957732, 5.43682005869899241867952100413, 6.45406179414174046853970226782, 7.22837361877626827819253873605, 8.286555927848319300094549099989, 9.010510832697850693854317813090
