| L(s) = 1 | + 3-s + 1.68·5-s − 2.74·7-s + 9-s − 2.18·11-s + 4.18·13-s + 1.68·15-s − 6.29·17-s − 4.93·19-s − 2.74·21-s − 3.44·23-s − 2.17·25-s + 27-s − 9.12·29-s + 0.616·31-s − 2.18·33-s − 4.61·35-s + 1.44·37-s + 4.18·39-s + 4.10·41-s − 8.53·43-s + 1.68·45-s + 6.48·47-s + 0.539·49-s − 6.29·51-s − 0.664·53-s − 3.68·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.751·5-s − 1.03·7-s + 0.333·9-s − 0.660·11-s + 1.16·13-s + 0.434·15-s − 1.52·17-s − 1.13·19-s − 0.599·21-s − 0.718·23-s − 0.434·25-s + 0.192·27-s − 1.69·29-s + 0.110·31-s − 0.381·33-s − 0.780·35-s + 0.237·37-s + 0.670·39-s + 0.641·41-s − 1.30·43-s + 0.250·45-s + 0.946·47-s + 0.0771·49-s − 0.881·51-s − 0.0913·53-s − 0.496·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2832 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| good | 5 | \( 1 - 1.68T + 5T^{2} \) |
| 7 | \( 1 + 2.74T + 7T^{2} \) |
| 11 | \( 1 + 2.18T + 11T^{2} \) |
| 13 | \( 1 - 4.18T + 13T^{2} \) |
| 17 | \( 1 + 6.29T + 17T^{2} \) |
| 19 | \( 1 + 4.93T + 19T^{2} \) |
| 23 | \( 1 + 3.44T + 23T^{2} \) |
| 29 | \( 1 + 9.12T + 29T^{2} \) |
| 31 | \( 1 - 0.616T + 31T^{2} \) |
| 37 | \( 1 - 1.44T + 37T^{2} \) |
| 41 | \( 1 - 4.10T + 41T^{2} \) |
| 43 | \( 1 + 8.53T + 43T^{2} \) |
| 47 | \( 1 - 6.48T + 47T^{2} \) |
| 53 | \( 1 + 0.664T + 53T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 - 9.93T + 67T^{2} \) |
| 71 | \( 1 + 6.82T + 71T^{2} \) |
| 73 | \( 1 - 13.5T + 73T^{2} \) |
| 79 | \( 1 - 1.17T + 79T^{2} \) |
| 83 | \( 1 - 15.3T + 83T^{2} \) |
| 89 | \( 1 - 1.97T + 89T^{2} \) |
| 97 | \( 1 - 3.33T + 97T^{2} \) |
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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
−8.485814971071825791263242058763, −7.79576474382844908263516865299, −6.66250887431661279031372286843, −6.28060759382446724887511886433, −5.47685947982184373706256412999, −4.24849519250029584635881358310, −3.59144151863374021969660379886, −2.49438483542100035042474985892, −1.83231473040240916365847419289, 0,
1.83231473040240916365847419289, 2.49438483542100035042474985892, 3.59144151863374021969660379886, 4.24849519250029584635881358310, 5.47685947982184373706256412999, 6.28060759382446724887511886433, 6.66250887431661279031372286843, 7.79576474382844908263516865299, 8.485814971071825791263242058763
