| L(s) = 1 | − 1.74·2-s − 0.00460·3-s + 1.03·4-s + 5-s + 0.00801·6-s + 7-s + 1.68·8-s − 2.99·9-s − 1.74·10-s − 1.64·11-s − 0.00474·12-s + 2.97·13-s − 1.74·14-s − 0.00460·15-s − 4.99·16-s − 6.18·17-s + 5.22·18-s − 2.19·19-s + 1.03·20-s − 0.00460·21-s + 2.86·22-s − 4.05·23-s − 0.00776·24-s + 25-s − 5.17·26-s + 0.0276·27-s + 1.03·28-s + ⋯ |
| L(s) = 1 | − 1.23·2-s − 0.00265·3-s + 0.515·4-s + 0.447·5-s + 0.00327·6-s + 0.377·7-s + 0.596·8-s − 0.999·9-s − 0.550·10-s − 0.496·11-s − 0.00137·12-s + 0.824·13-s − 0.465·14-s − 0.00118·15-s − 1.24·16-s − 1.49·17-s + 1.23·18-s − 0.504·19-s + 0.230·20-s − 0.00100·21-s + 0.610·22-s − 0.846·23-s − 0.00158·24-s + 0.200·25-s − 1.01·26-s + 0.00531·27-s + 0.194·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 229 | \( 1 + T \) |
| good | 2 | \( 1 + 1.74T + 2T^{2} \) |
| 3 | \( 1 + 0.00460T + 3T^{2} \) |
| 11 | \( 1 + 1.64T + 11T^{2} \) |
| 13 | \( 1 - 2.97T + 13T^{2} \) |
| 17 | \( 1 + 6.18T + 17T^{2} \) |
| 19 | \( 1 + 2.19T + 19T^{2} \) |
| 23 | \( 1 + 4.05T + 23T^{2} \) |
| 29 | \( 1 - 6.30T + 29T^{2} \) |
| 31 | \( 1 - 3.05T + 31T^{2} \) |
| 37 | \( 1 + 3.54T + 37T^{2} \) |
| 41 | \( 1 - 2.54T + 41T^{2} \) |
| 43 | \( 1 - 7.24T + 43T^{2} \) |
| 47 | \( 1 - 9.14T + 47T^{2} \) |
| 53 | \( 1 - 2.17T + 53T^{2} \) |
| 59 | \( 1 + 4.52T + 59T^{2} \) |
| 61 | \( 1 - 8.28T + 61T^{2} \) |
| 67 | \( 1 + 9.22T + 67T^{2} \) |
| 71 | \( 1 - 9.40T + 71T^{2} \) |
| 73 | \( 1 + 3.58T + 73T^{2} \) |
| 79 | \( 1 + 5.55T + 79T^{2} \) |
| 83 | \( 1 - 4.54T + 83T^{2} \) |
| 89 | \( 1 - 4.00T + 89T^{2} \) |
| 97 | \( 1 + 13.5T + 97T^{2} \) |
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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
−7.82521953268943445004959323096, −6.86736022397423921774258910957, −6.26174982582304696667801944893, −5.54823070983642198953142432949, −4.64787130597337727579633209452, −3.99565319595425730293287834076, −2.64139616812045311063075162344, −2.14977414326755202006120399109, −1.05360116176262212032264931453, 0,
1.05360116176262212032264931453, 2.14977414326755202006120399109, 2.64139616812045311063075162344, 3.99565319595425730293287834076, 4.64787130597337727579633209452, 5.54823070983642198953142432949, 6.26174982582304696667801944893, 6.86736022397423921774258910957, 7.82521953268943445004959323096
