| L(s) = 1 | + 1.63·2-s − 9.87·3-s − 5.33·4-s + 16.8·5-s − 16.1·6-s − 21.7·8-s + 70.6·9-s + 27.4·10-s − 8.55·11-s + 52.7·12-s + 11.3·13-s − 166.·15-s + 7.23·16-s − 68.4·17-s + 115.·18-s − 6.93·19-s − 89.8·20-s − 13.9·22-s − 132.·23-s + 214.·24-s + 157.·25-s + 18.5·26-s − 430.·27-s − 29·29-s − 271.·30-s + 0.419·31-s + 185.·32-s + ⋯ |
| L(s) = 1 | + 0.576·2-s − 1.90·3-s − 0.667·4-s + 1.50·5-s − 1.09·6-s − 0.961·8-s + 2.61·9-s + 0.867·10-s − 0.234·11-s + 1.26·12-s + 0.241·13-s − 2.86·15-s + 0.113·16-s − 0.976·17-s + 1.50·18-s − 0.0836·19-s − 1.00·20-s − 0.135·22-s − 1.19·23-s + 1.82·24-s + 1.26·25-s + 0.139·26-s − 3.07·27-s − 0.185·29-s − 1.64·30-s + 0.00242·31-s + 1.02·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1421 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1421 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 29 | \( 1 + 29T \) |
| good | 2 | \( 1 - 1.63T + 8T^{2} \) |
| 3 | \( 1 + 9.87T + 27T^{2} \) |
| 5 | \( 1 - 16.8T + 125T^{2} \) |
| 11 | \( 1 + 8.55T + 1.33e3T^{2} \) |
| 13 | \( 1 - 11.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 68.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 6.93T + 6.85e3T^{2} \) |
| 23 | \( 1 + 132.T + 1.21e4T^{2} \) |
| 31 | \( 1 - 0.419T + 2.97e4T^{2} \) |
| 37 | \( 1 - 395.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 447.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 184.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 97.2T + 1.03e5T^{2} \) |
| 53 | \( 1 + 209.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 45.9T + 2.05e5T^{2} \) |
| 61 | \( 1 + 427.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 405.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 557.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 381.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 577.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 353.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 277.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 677.T + 9.12e5T^{2} \) |
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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
−9.223907483131692478897589356387, −7.78491803991417583770279113757, −6.52526423598835651807374934903, −5.99305316515366395150046312104, −5.64394618447451274361239646533, −4.70398813080088189051034884935, −4.13371177924014846289952134281, −2.37115191886809887755433751136, −1.13095939722162695104817925737, 0,
1.13095939722162695104817925737, 2.37115191886809887755433751136, 4.13371177924014846289952134281, 4.70398813080088189051034884935, 5.64394618447451274361239646533, 5.99305316515366395150046312104, 6.52526423598835651807374934903, 7.78491803991417583770279113757, 9.223907483131692478897589356387
