| L(s) = 1 | − 0.626·2-s + 3·3-s − 7.60·4-s + 4.84·5-s − 1.87·6-s + 9.77·8-s + 9·9-s − 3.03·10-s − 11·11-s − 22.8·12-s − 28.0·13-s + 14.5·15-s + 54.7·16-s + 117.·17-s − 5.63·18-s − 38.9·19-s − 36.8·20-s + 6.88·22-s − 125.·23-s + 29.3·24-s − 101.·25-s + 17.5·26-s + 27·27-s + 108.·29-s − 9.11·30-s − 274.·31-s − 112.·32-s + ⋯ |
| L(s) = 1 | − 0.221·2-s + 0.577·3-s − 0.950·4-s + 0.433·5-s − 0.127·6-s + 0.431·8-s + 0.333·9-s − 0.0960·10-s − 0.301·11-s − 0.549·12-s − 0.597·13-s + 0.250·15-s + 0.855·16-s + 1.67·17-s − 0.0738·18-s − 0.470·19-s − 0.412·20-s + 0.0667·22-s − 1.13·23-s + 0.249·24-s − 0.811·25-s + 0.132·26-s + 0.192·27-s + 0.696·29-s − 0.0554·30-s − 1.58·31-s − 0.621·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - 3T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 + 0.626T + 8T^{2} \) |
| 5 | \( 1 - 4.84T + 125T^{2} \) |
| 13 | \( 1 + 28.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 117.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 38.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 125.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 108.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 274.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 257.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 178.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 169.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 342.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 249.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 367.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 309.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 732.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 683.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 589.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 929.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 561.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.17e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.08e3T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.671162126927640240561205694528, −7.86908317422024080111650579835, −7.43679235132524504826644848048, −6.00580402281566347051235706907, −5.37323696754431583655310102647, −4.34531813507511808768335084352, −3.56360000163468999634928944855, −2.42304849285641513514818016302, −1.30185839663816882557466990766, 0,
1.30185839663816882557466990766, 2.42304849285641513514818016302, 3.56360000163468999634928944855, 4.34531813507511808768335084352, 5.37323696754431583655310102647, 6.00580402281566347051235706907, 7.43679235132524504826644848048, 7.86908317422024080111650579835, 8.671162126927640240561205694528
