| L(s) = 1 | − 21.4·2-s − 17.3·3-s + 330.·4-s − 555.·5-s + 371.·6-s − 1.11e3·7-s − 4.33e3·8-s − 1.88e3·9-s + 1.19e4·10-s − 560.·11-s − 5.73e3·12-s − 492.·13-s + 2.38e4·14-s + 9.65e3·15-s + 5.04e4·16-s − 519.·17-s + 4.03e4·18-s − 50.3·19-s − 1.83e5·20-s + 1.93e4·21-s + 1.19e4·22-s − 7.52e4·23-s + 7.52e4·24-s + 2.30e5·25-s + 1.05e4·26-s + 7.07e4·27-s − 3.67e5·28-s + ⋯ |
| L(s) = 1 | − 1.89·2-s − 0.371·3-s + 2.58·4-s − 1.98·5-s + 0.702·6-s − 1.22·7-s − 2.99·8-s − 0.862·9-s + 3.76·10-s − 0.126·11-s − 0.958·12-s − 0.0621·13-s + 2.32·14-s + 0.738·15-s + 3.07·16-s − 0.0256·17-s + 1.63·18-s − 0.00168·19-s − 5.13·20-s + 0.455·21-s + 0.240·22-s − 1.28·23-s + 1.11·24-s + 2.95·25-s + 0.117·26-s + 0.691·27-s − 3.16·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.08246292084\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08246292084\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + 2.43e4T \) |
| good | 2 | \( 1 + 21.4T + 128T^{2} \) |
| 3 | \( 1 + 17.3T + 2.18e3T^{2} \) |
| 5 | \( 1 + 555.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.11e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 560.T + 1.94e7T^{2} \) |
| 13 | \( 1 + 492.T + 6.27e7T^{2} \) |
| 17 | \( 1 + 519.T + 4.10e8T^{2} \) |
| 19 | \( 1 + 50.3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 7.52e4T + 3.40e9T^{2} \) |
| 31 | \( 1 - 8.42e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.32e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 1.81e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 4.83e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.11e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.04e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.26e4T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.61e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.72e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.88e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.05e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 5.94e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 3.87e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 3.84e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 5.95e6T + 8.07e13T^{2} \) |
| 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
−16.12341978932238950005518151430, −15.14648034705356490938183855427, −12.20771101646723274415645583638, −11.49626809680167292475813228034, −10.31240379070344681009221434655, −8.771852880226940741507750324078, −7.77823570871472461150797007931, −6.53594824290453757442663399233, −3.20696783786290746092773415284, −0.29454409729643380045860049541,
0.29454409729643380045860049541, 3.20696783786290746092773415284, 6.53594824290453757442663399233, 7.77823570871472461150797007931, 8.771852880226940741507750324078, 10.31240379070344681009221434655, 11.49626809680167292475813228034, 12.20771101646723274415645583638, 15.14648034705356490938183855427, 16.12341978932238950005518151430
