| L(s) = 1 | − 3.56·2-s − 14.4·3-s − 19.3·4-s − 25·5-s + 51.5·6-s − 241.·7-s + 182.·8-s − 33.1·9-s + 89.0·10-s − 121·11-s + 279.·12-s + 906.·13-s + 858.·14-s + 362.·15-s − 32.9·16-s + 845.·17-s + 118.·18-s + 361·19-s + 482.·20-s + 3.49e3·21-s + 430.·22-s + 4.06e3·23-s − 2.64e3·24-s + 625·25-s − 3.22e3·26-s + 4.00e3·27-s + 4.65e3·28-s + ⋯ |
| L(s) = 1 | − 0.629·2-s − 0.929·3-s − 0.603·4-s − 0.447·5-s + 0.585·6-s − 1.85·7-s + 1.00·8-s − 0.136·9-s + 0.281·10-s − 0.301·11-s + 0.560·12-s + 1.48·13-s + 1.17·14-s + 0.415·15-s − 0.0322·16-s + 0.709·17-s + 0.0860·18-s + 0.229·19-s + 0.269·20-s + 1.72·21-s + 0.189·22-s + 1.60·23-s − 0.938·24-s + 0.200·25-s − 0.936·26-s + 1.05·27-s + 1.12·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.2948385623\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2948385623\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 11 | \( 1 + 121T \) |
| 19 | \( 1 - 361T \) |
| good | 2 | \( 1 + 3.56T + 32T^{2} \) |
| 3 | \( 1 + 14.4T + 243T^{2} \) |
| 7 | \( 1 + 241.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 906.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 845.T + 1.41e6T^{2} \) |
| 23 | \( 1 - 4.06e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.49e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.54e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 827.T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.15e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.77e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.20e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.13e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.04e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 7.22e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.71e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.22e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 8.79e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.02e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.34e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.18e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 9.05e4T + 8.58e9T^{2} \) |
| 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
−9.106468995322069967144254105765, −8.656113960697991939682477413155, −7.47763503770780170465499076688, −6.66923308013808644480314853032, −5.81702104026558201348727597094, −5.08113435900804995276638770673, −3.71796747461754137134398426985, −3.18150901667107509763200363099, −1.18233591381317413737970251372, −0.32327046854726625441314071452,
0.32327046854726625441314071452, 1.18233591381317413737970251372, 3.18150901667107509763200363099, 3.71796747461754137134398426985, 5.08113435900804995276638770673, 5.81702104026558201348727597094, 6.66923308013808644480314853032, 7.47763503770780170465499076688, 8.656113960697991939682477413155, 9.106468995322069967144254105765
