| L(s) = 1 | + 2·2-s − 6.27·3-s + 4·4-s + 2.82·5-s − 12.5·6-s − 7·7-s + 8·8-s + 12.3·9-s + 5.64·10-s − 11·11-s − 25.0·12-s − 50.5·13-s − 14·14-s − 17.7·15-s + 16·16-s − 131.·17-s + 24.7·18-s − 52.9·19-s + 11.2·20-s + 43.9·21-s − 22·22-s + 87.7·23-s − 50.1·24-s − 117.·25-s − 101.·26-s + 91.7·27-s − 28·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.20·3-s + 0.5·4-s + 0.252·5-s − 0.853·6-s − 0.377·7-s + 0.353·8-s + 0.458·9-s + 0.178·10-s − 0.301·11-s − 0.603·12-s − 1.07·13-s − 0.267·14-s − 0.305·15-s + 0.250·16-s − 1.87·17-s + 0.324·18-s − 0.638·19-s + 0.126·20-s + 0.456·21-s − 0.213·22-s + 0.795·23-s − 0.426·24-s − 0.936·25-s − 0.762·26-s + 0.654·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - 2T \) |
| 7 | \( 1 + 7T \) |
| 11 | \( 1 + 11T \) |
| good | 3 | \( 1 + 6.27T + 27T^{2} \) |
| 5 | \( 1 - 2.82T + 125T^{2} \) |
| 13 | \( 1 + 50.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 131.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 52.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 87.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + 119.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 1.52T + 2.97e4T^{2} \) |
| 37 | \( 1 + 3.32T + 5.06e4T^{2} \) |
| 41 | \( 1 + 135.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 105.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 506.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 725.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 605.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 504.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 630.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 754.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 111.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 609.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 901.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 268.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 95.1T + 9.12e5T^{2} \) |
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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
−12.04791050416748331492745565839, −11.12966572868641268270197757556, −10.37088480633992929004825734677, −9.027897301482134370732956435305, −7.26217152806617221887042633306, −6.33954231512051012233379021006, −5.37210849450151260389785771841, −4.33632725772534230159865794849, −2.39840930516018404401566265642, 0,
2.39840930516018404401566265642, 4.33632725772534230159865794849, 5.37210849450151260389785771841, 6.33954231512051012233379021006, 7.26217152806617221887042633306, 9.027897301482134370732956435305, 10.37088480633992929004825734677, 11.12966572868641268270197757556, 12.04791050416748331492745565839
