| L(s) = 1 | − 3.29e5·5-s + 4.30e7·9-s + 1.63e9·13-s − 9.93e9·17-s − 4.39e10·25-s − 9.81e11·29-s + 6.16e12·37-s − 3.16e12·41-s − 1.41e13·45-s + 3.32e13·49-s + 3.19e13·53-s − 4.59e13·61-s − 5.37e14·65-s + 1.38e15·73-s + 1.85e15·81-s + 3.27e15·85-s − 6.95e15·89-s + 1.43e16·97-s + 5.17e14·101-s + 1.95e15·109-s + 2.57e16·113-s + 7.02e16·117-s + ⋯ |
| L(s) = 1 | − 0.843·5-s + 9-s + 1.99·13-s − 1.42·17-s − 0.287·25-s − 1.96·29-s + 1.75·37-s − 0.396·41-s − 0.843·45-s + 49-s + 0.513·53-s − 0.239·61-s − 1.68·65-s + 1.71·73-s + 81-s + 1.20·85-s − 1.76·89-s + 1.83·97-s + 0.0477·101-s + 0.0981·109-s + 0.968·113-s + 1.99·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{17}{2})\) |
\(\approx\) |
\(1.943770340\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.943770340\) |
| \(L(9)\) |
|
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 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 5 | \( 1 + 329666 T + p^{16} T^{2} \) |
| 7 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 11 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 13 | \( 1 - 1631232958 T + p^{16} T^{2} \) |
| 17 | \( 1 + 9937278718 T + p^{16} T^{2} \) |
| 19 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 23 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 29 | \( 1 + 981515008322 T + p^{16} T^{2} \) |
| 31 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 37 | \( 1 - 6167627357758 T + p^{16} T^{2} \) |
| 41 | \( 1 + 3168324620158 T + p^{16} T^{2} \) |
| 43 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 47 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 53 | \( 1 - 31962705295678 T + p^{16} T^{2} \) |
| 59 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 61 | \( 1 + 45990056420162 T + p^{16} T^{2} \) |
| 67 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 71 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 73 | \( 1 - 1381042818437762 T + p^{16} T^{2} \) |
| 79 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 83 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 89 | \( 1 + 6957151819021438 T + p^{16} T^{2} \) |
| 97 | \( 1 - 14385701036152322 T + p^{16} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.46933740811911256481236938452, −10.83021914430229935323919140496, −9.336987936968504755387192262068, −8.257811325277540397298723651705, −7.15708822334191752667032511563, −5.99825822565118668293775380896, −4.31012235174103654154330420909, −3.66188229134996104351411410265, −1.90103096908713734240171442078, −0.68328176224705532529351481045,
0.68328176224705532529351481045, 1.90103096908713734240171442078, 3.66188229134996104351411410265, 4.31012235174103654154330420909, 5.99825822565118668293775380896, 7.15708822334191752667032511563, 8.257811325277540397298723651705, 9.336987936968504755387192262068, 10.83021914430229935323919140496, 11.46933740811911256481236938452
