| L(s) = 1 | − 256·2-s + 6.55e4·4-s − 3.29e5·5-s − 1.67e7·8-s + 8.43e7·10-s − 1.63e9·13-s + 4.29e9·16-s + 9.93e9·17-s − 2.16e10·20-s − 4.39e10·25-s + 4.17e11·26-s − 9.81e11·29-s − 1.09e12·32-s − 2.54e12·34-s − 6.16e12·37-s + 5.53e12·40-s + 3.16e12·41-s + 3.32e13·49-s + 1.12e13·50-s − 1.06e14·52-s + 3.19e13·53-s + 2.51e14·58-s + 4.59e13·61-s + 2.81e14·64-s + 5.37e14·65-s + 6.51e14·68-s + 1.38e15·73-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s − 0.843·5-s − 8-s + 0.843·10-s − 1.99·13-s + 16-s + 1.42·17-s − 0.843·20-s − 0.287·25-s + 1.99·26-s − 1.96·29-s − 32-s − 1.42·34-s − 1.75·37-s + 0.843·40-s + 0.396·41-s + 49-s + 0.287·50-s − 1.99·52-s + 0.513·53-s + 1.96·58-s + 0.239·61-s + 64-s + 1.68·65-s + 1.42·68-s + 1.71·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{17}{2})\) |
\(\approx\) |
\(0.6146850728\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6146850728\) |
| \(L(9)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{8} T \) |
| 3 | \( 1 \) |
| good | 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
−12.41383291864221722526867009603, −11.72495969361213974357527610974, −10.33686191403028841180694581420, −9.322527895642277904452969807312, −7.82394053707413942107252954254, −7.21906029882322626964171707383, −5.39186411672546395342592269537, −3.52200050178642465308797174810, −2.08974116434938036454140535465, −0.47351017362565416962240245692,
0.47351017362565416962240245692, 2.08974116434938036454140535465, 3.52200050178642465308797174810, 5.39186411672546395342592269537, 7.21906029882322626964171707383, 7.82394053707413942107252954254, 9.322527895642277904452969807312, 10.33686191403028841180694581420, 11.72495969361213974357527610974, 12.41383291864221722526867009603
