| L(s) = 1 | + 21.9·3-s + 184.·5-s + 1.05e3·7-s − 1.70e3·9-s + 4.32e3·11-s − 1.12e4·13-s + 4.05e3·15-s − 2.17e4·17-s − 4.54e4·19-s + 2.30e4·21-s + 4.41e3·23-s − 4.39e4·25-s − 8.54e4·27-s + 2.36e4·29-s − 7.29e4·31-s + 9.48e4·33-s + 1.94e5·35-s − 4.83e5·37-s − 2.46e5·39-s + 4.11e5·41-s − 9.61e4·43-s − 3.15e5·45-s + 1.56e5·47-s + 2.83e5·49-s − 4.77e5·51-s + 6.86e5·53-s + 7.99e5·55-s + ⋯ |
| L(s) = 1 | + 0.469·3-s + 0.661·5-s + 1.15·7-s − 0.779·9-s + 0.979·11-s − 1.42·13-s + 0.310·15-s − 1.07·17-s − 1.52·19-s + 0.543·21-s + 0.0756·23-s − 0.562·25-s − 0.835·27-s + 0.180·29-s − 0.439·31-s + 0.459·33-s + 0.766·35-s − 1.56·37-s − 0.666·39-s + 0.931·41-s − 0.184·43-s − 0.515·45-s + 0.219·47-s + 0.343·49-s − 0.503·51-s + 0.633·53-s + 0.648·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 3 | \( 1 - 21.9T + 2.18e3T^{2} \) |
| 5 | \( 1 - 184.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.05e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 4.32e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.12e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.17e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 4.54e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 4.41e3T + 3.40e9T^{2} \) |
| 29 | \( 1 - 2.36e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 7.29e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 4.83e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 4.11e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 9.61e4T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.56e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 6.86e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.79e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.36e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.08e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 5.60e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.16e4T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.34e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 8.82e5T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.34e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 7.32e6T + 8.07e13T^{2} \) |
| 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
−10.27924237727460953751428506259, −9.098804968395960242787061202343, −8.569355793720304683909318133255, −7.39992476849732772984978545100, −6.26759518233967098930002757195, −5.10780824228407406852775906248, −4.09260062729536611931909951576, −2.43223111500508726904436361730, −1.78552759647879903164172717419, 0,
1.78552759647879903164172717419, 2.43223111500508726904436361730, 4.09260062729536611931909951576, 5.10780824228407406852775906248, 6.26759518233967098930002757195, 7.39992476849732772984978545100, 8.569355793720304683909318133255, 9.098804968395960242787061202343, 10.27924237727460953751428506259
