| L(s) = 1 | − 0.632·2-s + 0.799·3-s − 1.59·4-s − 5-s − 0.506·6-s − 7-s + 2.27·8-s − 2.36·9-s + 0.632·10-s + 1.65·11-s − 1.27·12-s − 2.61·13-s + 0.632·14-s − 0.799·15-s + 1.75·16-s + 3.33·17-s + 1.49·18-s − 0.169·19-s + 1.59·20-s − 0.799·21-s − 1.04·22-s − 4.56·23-s + 1.82·24-s + 25-s + 1.65·26-s − 4.28·27-s + 1.59·28-s + ⋯ |
| L(s) = 1 | − 0.447·2-s + 0.461·3-s − 0.799·4-s − 0.447·5-s − 0.206·6-s − 0.377·7-s + 0.805·8-s − 0.786·9-s + 0.200·10-s + 0.500·11-s − 0.369·12-s − 0.724·13-s + 0.169·14-s − 0.206·15-s + 0.439·16-s + 0.808·17-s + 0.351·18-s − 0.0387·19-s + 0.357·20-s − 0.174·21-s − 0.223·22-s − 0.951·23-s + 0.371·24-s + 0.200·25-s + 0.324·26-s − 0.825·27-s + 0.302·28-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 | \( 1 + 0.632T + 2T^{2} \) |
| 3 | \( 1 - 0.799T + 3T^{2} \) |
| 11 | \( 1 - 1.65T + 11T^{2} \) |
| 13 | \( 1 + 2.61T + 13T^{2} \) |
| 17 | \( 1 - 3.33T + 17T^{2} \) |
| 19 | \( 1 + 0.169T + 19T^{2} \) |
| 23 | \( 1 + 4.56T + 23T^{2} \) |
| 29 | \( 1 - 6.56T + 29T^{2} \) |
| 31 | \( 1 - 5.60T + 31T^{2} \) |
| 37 | \( 1 + 0.703T + 37T^{2} \) |
| 41 | \( 1 + 6.52T + 41T^{2} \) |
| 43 | \( 1 + 2.26T + 43T^{2} \) |
| 47 | \( 1 - 3.91T + 47T^{2} \) |
| 53 | \( 1 + 9.68T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 - 10.8T + 61T^{2} \) |
| 67 | \( 1 + 4.45T + 67T^{2} \) |
| 71 | \( 1 - 7.65T + 71T^{2} \) |
| 73 | \( 1 - 1.41T + 73T^{2} \) |
| 79 | \( 1 + 12.8T + 79T^{2} \) |
| 83 | \( 1 + 0.812T + 83T^{2} \) |
| 89 | \( 1 - 11.0T + 89T^{2} \) |
| 97 | \( 1 - 2.59T + 97T^{2} \) |
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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
−7.76188149313697390982582589497, −6.95319233836874263319185935318, −6.14907218513489427835379748932, −5.30316029926478459587517788496, −4.62229012199002824940051102050, −3.80029645611792181338949906301, −3.20295787403452155868734284599, −2.26745986129748234737294419977, −1.01580619806044142777985470938, 0,
1.01580619806044142777985470938, 2.26745986129748234737294419977, 3.20295787403452155868734284599, 3.80029645611792181338949906301, 4.62229012199002824940051102050, 5.30316029926478459587517788496, 6.14907218513489427835379748932, 6.95319233836874263319185935318, 7.76188149313697390982582589497
