| L(s) = 1 | + 8·2-s + 64·4-s + 185.·7-s + 512·8-s − 3.56e3·11-s + 6.09e3·13-s + 1.48e3·14-s + 4.09e3·16-s + 1.24e4·17-s − 5.06e4·19-s − 2.84e4·22-s − 1.14e4·23-s + 4.87e4·26-s + 1.18e4·28-s − 1.01e5·29-s − 2.90e4·31-s + 3.27e4·32-s + 9.97e4·34-s + 1.49e5·37-s − 4.05e5·38-s + 3.74e5·41-s − 1.74e5·43-s − 2.27e5·44-s − 9.15e4·46-s + 4.28e5·47-s − 7.89e5·49-s + 3.90e5·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.204·7-s + 0.353·8-s − 0.806·11-s + 0.769·13-s + 0.144·14-s + 0.250·16-s + 0.615·17-s − 1.69·19-s − 0.570·22-s − 0.196·23-s + 0.544·26-s + 0.102·28-s − 0.776·29-s − 0.175·31-s + 0.176·32-s + 0.435·34-s + 0.484·37-s − 1.19·38-s + 0.848·41-s − 0.334·43-s − 0.403·44-s − 0.138·46-s + 0.601·47-s − 0.958·49-s + 0.384·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 8T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 185.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 3.56e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 6.09e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.24e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 5.06e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 1.14e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.01e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.90e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.49e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 3.74e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 1.74e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 4.28e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.71e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.34e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.39e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.60e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.91e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.19e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 4.70e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 9.19e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 6.43e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.26e7T + 8.07e13T^{2} \) |
| 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
−9.561215876394600424421358741446, −8.374146497875781960296678004949, −7.66252878688094910152667771158, −6.45860136924397388092003986044, −5.70054225728703442163803236721, −4.65379503698302706851937841161, −3.71321474091213529429111589498, −2.57903430492260954369627349893, −1.50053354951703354754029326458, 0,
1.50053354951703354754029326458, 2.57903430492260954369627349893, 3.71321474091213529429111589498, 4.65379503698302706851937841161, 5.70054225728703442163803236721, 6.45860136924397388092003986044, 7.66252878688094910152667771158, 8.374146497875781960296678004949, 9.561215876394600424421358741446
