| L(s) = 1 | + 142.·2-s + 2.56e3·3-s − 1.23e4·4-s + 2.82e5·5-s + 3.66e5·6-s − 9.61e5·7-s − 6.44e6·8-s − 7.76e6·9-s + 4.04e7·10-s − 1.23e8·11-s − 3.16e7·12-s − 3.18e8·13-s − 1.37e8·14-s + 7.26e8·15-s − 5.17e8·16-s + 2.95e9·17-s − 1.10e9·18-s − 4.34e8·19-s − 3.48e9·20-s − 2.46e9·21-s − 1.76e10·22-s − 1.46e10·23-s − 1.65e10·24-s + 4.95e10·25-s − 4.55e10·26-s − 5.67e10·27-s + 1.18e10·28-s + ⋯ |
| L(s) = 1 | + 0.789·2-s + 0.677·3-s − 0.376·4-s + 1.61·5-s + 0.535·6-s − 0.441·7-s − 1.08·8-s − 0.541·9-s + 1.27·10-s − 1.90·11-s − 0.254·12-s − 1.40·13-s − 0.348·14-s + 1.09·15-s − 0.482·16-s + 1.74·17-s − 0.427·18-s − 0.111·19-s − 0.609·20-s − 0.298·21-s − 1.50·22-s − 0.894·23-s − 0.736·24-s + 1.62·25-s − 1.11·26-s − 1.04·27-s + 0.165·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 - 1.72e10T \) |
| good | 2 | \( 1 - 142.T + 3.27e4T^{2} \) |
| 3 | \( 1 - 2.56e3T + 1.43e7T^{2} \) |
| 5 | \( 1 - 2.82e5T + 3.05e10T^{2} \) |
| 7 | \( 1 + 9.61e5T + 4.74e12T^{2} \) |
| 11 | \( 1 + 1.23e8T + 4.17e15T^{2} \) |
| 13 | \( 1 + 3.18e8T + 5.11e16T^{2} \) |
| 17 | \( 1 - 2.95e9T + 2.86e18T^{2} \) |
| 19 | \( 1 + 4.34e8T + 1.51e19T^{2} \) |
| 23 | \( 1 + 1.46e10T + 2.66e20T^{2} \) |
| 31 | \( 1 + 2.00e11T + 2.34e22T^{2} \) |
| 37 | \( 1 - 6.88e11T + 3.33e23T^{2} \) |
| 41 | \( 1 - 5.49e10T + 1.55e24T^{2} \) |
| 43 | \( 1 + 1.94e12T + 3.17e24T^{2} \) |
| 47 | \( 1 - 3.02e12T + 1.20e25T^{2} \) |
| 53 | \( 1 + 8.85e11T + 7.31e25T^{2} \) |
| 59 | \( 1 + 1.78e12T + 3.65e26T^{2} \) |
| 61 | \( 1 + 7.63e12T + 6.02e26T^{2} \) |
| 67 | \( 1 + 2.44e13T + 2.46e27T^{2} \) |
| 71 | \( 1 - 1.02e13T + 5.87e27T^{2} \) |
| 73 | \( 1 - 5.92e13T + 8.90e27T^{2} \) |
| 79 | \( 1 - 1.17e14T + 2.91e28T^{2} \) |
| 83 | \( 1 + 3.59e14T + 6.11e28T^{2} \) |
| 89 | \( 1 - 4.69e14T + 1.74e29T^{2} \) |
| 97 | \( 1 - 7.41e14T + 6.33e29T^{2} \) |
| 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
−13.28995827242351972755862058136, −12.46333072346030781376827277738, −10.12626304956917897910031838483, −9.463597258376468707194300519111, −7.898707120871578923503140298841, −5.80066317589644123308967856126, −5.18219210872459671799752761641, −3.08899776295517600569947605990, −2.31510398884108638333413333938, 0,
2.31510398884108638333413333938, 3.08899776295517600569947605990, 5.18219210872459671799752761641, 5.80066317589644123308967856126, 7.898707120871578923503140298841, 9.463597258376468707194300519111, 10.12626304956917897910031838483, 12.46333072346030781376827277738, 13.28995827242351972755862058136
