| L(s) = 1 | + 1.37·2-s + 3-s − 0.103·4-s + 1.10·5-s + 1.37·6-s − 7-s − 2.89·8-s − 2·9-s + 1.51·10-s − 2.10·11-s − 0.103·12-s − 3.37·13-s − 1.37·14-s + 1.10·15-s − 3.78·16-s − 3.54·17-s − 2.75·18-s − 1.75·19-s − 0.113·20-s − 21-s − 2.89·22-s − 2.48·23-s − 2.89·24-s − 3.78·25-s − 4.65·26-s − 5·27-s + 0.103·28-s + ⋯ |
| L(s) = 1 | + 0.973·2-s + 0.577·3-s − 0.0516·4-s + 0.493·5-s + 0.562·6-s − 0.377·7-s − 1.02·8-s − 0.666·9-s + 0.480·10-s − 0.634·11-s − 0.0298·12-s − 0.936·13-s − 0.368·14-s + 0.284·15-s − 0.945·16-s − 0.860·17-s − 0.649·18-s − 0.402·19-s − 0.0254·20-s − 0.218·21-s − 0.617·22-s − 0.517·23-s − 0.591·24-s − 0.756·25-s − 0.912·26-s − 0.962·27-s + 0.0195·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1603 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1603 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 + T \) |
| 229 | \( 1 + T \) |
| good | 2 | \( 1 - 1.37T + 2T^{2} \) |
| 3 | \( 1 - T + 3T^{2} \) |
| 5 | \( 1 - 1.10T + 5T^{2} \) |
| 11 | \( 1 + 2.10T + 11T^{2} \) |
| 13 | \( 1 + 3.37T + 13T^{2} \) |
| 17 | \( 1 + 3.54T + 17T^{2} \) |
| 19 | \( 1 + 1.75T + 19T^{2} \) |
| 23 | \( 1 + 2.48T + 23T^{2} \) |
| 29 | \( 1 - 2.09T + 29T^{2} \) |
| 31 | \( 1 - 5.54T + 31T^{2} \) |
| 37 | \( 1 - 5.67T + 37T^{2} \) |
| 41 | \( 1 + 4.65T + 41T^{2} \) |
| 43 | \( 1 - 8.88T + 43T^{2} \) |
| 47 | \( 1 + 0.904T + 47T^{2} \) |
| 53 | \( 1 - 11.4T + 53T^{2} \) |
| 59 | \( 1 + 4.09T + 59T^{2} \) |
| 61 | \( 1 - 2.84T + 61T^{2} \) |
| 67 | \( 1 + 1.54T + 67T^{2} \) |
| 71 | \( 1 + 13.2T + 71T^{2} \) |
| 73 | \( 1 + 9.35T + 73T^{2} \) |
| 79 | \( 1 + 16.6T + 79T^{2} \) |
| 83 | \( 1 - 8.19T + 83T^{2} \) |
| 89 | \( 1 - 5.92T + 89T^{2} \) |
| 97 | \( 1 + 13.4T + 97T^{2} \) |
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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
−8.991536155464484253046301145680, −8.336571163910664292245151535837, −7.36603826145964981366410241165, −6.23034333591364158330624685143, −5.72536290917748573367818597450, −4.75634934316269045388378732385, −3.98883072460153374691941725875, −2.80471907115419240200221067106, −2.37326388400154453625160256586, 0,
2.37326388400154453625160256586, 2.80471907115419240200221067106, 3.98883072460153374691941725875, 4.75634934316269045388378732385, 5.72536290917748573367818597450, 6.23034333591364158330624685143, 7.36603826145964981366410241165, 8.336571163910664292245151535837, 8.991536155464484253046301145680
