| L(s) = 1 | + 1.05e3·5-s + 6.56e3·9-s + 478·13-s − 6.33e4·17-s + 7.20e5·25-s + 1.40e6·29-s − 9.25e5·37-s + 3.57e6·41-s + 6.91e6·45-s + 5.76e6·49-s + 9.62e6·53-s − 2.07e7·61-s + 5.03e5·65-s − 5.47e7·73-s + 4.30e7·81-s − 6.67e7·85-s − 3.02e7·89-s − 1.73e8·97-s − 1.45e8·101-s + 1.94e8·109-s + 2.80e8·113-s + 3.13e6·117-s + ⋯ |
| L(s) = 1 | + 1.68·5-s + 9-s + 0.0167·13-s − 0.758·17-s + 1.84·25-s + 1.99·29-s − 0.494·37-s + 1.26·41-s + 1.68·45-s + 49-s + 1.21·53-s − 1.49·61-s + 0.0282·65-s − 1.92·73-s + 81-s − 1.27·85-s − 0.482·89-s − 1.95·97-s − 1.39·101-s + 1.37·109-s + 1.72·113-s + 0.0167·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.899080654\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.899080654\) |
| \(L(5)\) |
|
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 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 5 | \( 1 - 1054 T + p^{8} T^{2} \) |
| 7 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 11 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 13 | \( 1 - 478 T + p^{8} T^{2} \) |
| 17 | \( 1 + 63358 T + p^{8} T^{2} \) |
| 19 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 23 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 29 | \( 1 - 1407838 T + p^{8} T^{2} \) |
| 31 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 37 | \( 1 + 925922 T + p^{8} T^{2} \) |
| 41 | \( 1 - 3577922 T + p^{8} T^{2} \) |
| 43 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 47 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 53 | \( 1 - 9620638 T + p^{8} T^{2} \) |
| 59 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 61 | \( 1 + 20722082 T + p^{8} T^{2} \) |
| 67 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 71 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 73 | \( 1 + 54717118 T + p^{8} T^{2} \) |
| 79 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 83 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 89 | \( 1 + 30265918 T + p^{8} T^{2} \) |
| 97 | \( 1 + 173379838 T + p^{8} T^{2} \) |
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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
−13.36891026716447278461308621343, −12.36180730251515258308409325002, −10.63625746921863131145246209425, −9.868300755889128379745637285794, −8.809452365836967645155984374502, −7.00939779527830136639091926496, −5.94763205510150093144011033501, −4.56452906552356932160328903228, −2.49008690981722803803778949166, −1.24796396559190230129418665306,
1.24796396559190230129418665306, 2.49008690981722803803778949166, 4.56452906552356932160328903228, 5.94763205510150093144011033501, 7.00939779527830136639091926496, 8.809452365836967645155984374502, 9.868300755889128379745637285794, 10.63625746921863131145246209425, 12.36180730251515258308409325002, 13.36891026716447278461308621343
