| L(s) = 1 | − 8·2-s + 64·4-s + 165·5-s − 508·7-s − 512·8-s − 1.32e3·10-s − 3.02e3·11-s + 5.03e3·13-s + 4.06e3·14-s + 4.09e3·16-s + 3.18e3·17-s + 1.50e3·19-s + 1.05e4·20-s + 2.41e4·22-s + 7.56e4·23-s − 5.09e4·25-s − 4.03e4·26-s − 3.25e4·28-s + 8.26e4·29-s − 1.74e5·31-s − 3.27e4·32-s − 2.55e4·34-s − 8.38e4·35-s − 3.23e5·37-s − 1.20e4·38-s − 8.44e4·40-s + 3.08e5·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.590·5-s − 0.559·7-s − 0.353·8-s − 0.417·10-s − 0.685·11-s + 0.636·13-s + 0.395·14-s + 1/4·16-s + 0.157·17-s + 0.0504·19-s + 0.295·20-s + 0.484·22-s + 1.29·23-s − 0.651·25-s − 0.449·26-s − 0.279·28-s + 0.629·29-s − 1.05·31-s − 0.176·32-s − 0.111·34-s − 0.330·35-s − 1.05·37-s − 0.0356·38-s − 0.208·40-s + 0.698·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.420562673\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.420562673\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{3} T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 33 p T + p^{7} T^{2} \) |
| 7 | \( 1 + 508 T + p^{7} T^{2} \) |
| 11 | \( 1 + 3024 T + p^{7} T^{2} \) |
| 13 | \( 1 - 5039 T + p^{7} T^{2} \) |
| 17 | \( 1 - 3189 T + p^{7} T^{2} \) |
| 19 | \( 1 - 1508 T + p^{7} T^{2} \) |
| 23 | \( 1 - 75600 T + p^{7} T^{2} \) |
| 29 | \( 1 - 82665 T + p^{7} T^{2} \) |
| 31 | \( 1 + 174892 T + p^{7} T^{2} \) |
| 37 | \( 1 + 323569 T + p^{7} T^{2} \) |
| 41 | \( 1 - 308118 T + p^{7} T^{2} \) |
| 43 | \( 1 - 336680 T + p^{7} T^{2} \) |
| 47 | \( 1 - 383196 T + p^{7} T^{2} \) |
| 53 | \( 1 + 760206 T + p^{7} T^{2} \) |
| 59 | \( 1 - 2225664 T + p^{7} T^{2} \) |
| 61 | \( 1 - 2244815 T + p^{7} T^{2} \) |
| 67 | \( 1 - 1473188 T + p^{7} T^{2} \) |
| 71 | \( 1 - 5006892 T + p^{7} T^{2} \) |
| 73 | \( 1 + 5898301 T + p^{7} T^{2} \) |
| 79 | \( 1 - 7028768 T + p^{7} T^{2} \) |
| 83 | \( 1 - 2651196 T + p^{7} T^{2} \) |
| 89 | \( 1 - 6770901 T + p^{7} T^{2} \) |
| 97 | \( 1 - 16176386 T + p^{7} T^{2} \) |
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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
−11.27121190862890214399060433075, −10.41390497604905468566831904534, −9.521022211319110619683944604419, −8.628944445581923731139193937099, −7.42642812979380193590689330434, −6.34631407208950333116419438056, −5.28366390603081500853263125856, −3.43394493779649865361903169250, −2.15609466215075861131496236120, −0.73530117773738863068457326807,
0.73530117773738863068457326807, 2.15609466215075861131496236120, 3.43394493779649865361903169250, 5.28366390603081500853263125856, 6.34631407208950333116419438056, 7.42642812979380193590689330434, 8.628944445581923731139193937099, 9.521022211319110619683944604419, 10.41390497604905468566831904534, 11.27121190862890214399060433075
