| L(s) = 1 | + 64·4-s + 198·5-s + 343·7-s + 729·9-s − 2.19e3·13-s + 4.09e3·16-s − 1.14e4·19-s + 1.26e4·20-s − 1.97e4·23-s + 2.35e4·25-s + 2.19e4·28-s + 4.73e4·29-s − 7.81e3·31-s + 6.79e4·35-s + 4.66e4·36-s − 1.24e5·41-s + 2.76e4·43-s + 1.44e5·45-s − 5.34e3·47-s + 1.17e5·49-s − 1.40e5·52-s − 1.23e5·53-s + 2.10e5·59-s + 2.50e5·63-s + 2.62e5·64-s − 4.35e5·65-s − 4.39e5·73-s + ⋯ |
| L(s) = 1 | + 4-s + 1.58·5-s + 7-s + 9-s − 13-s + 16-s − 1.66·19-s + 1.58·20-s − 1.61·23-s + 1.50·25-s + 28-s + 1.94·29-s − 0.262·31-s + 1.58·35-s + 36-s − 1.80·41-s + 0.347·43-s + 1.58·45-s − 0.0514·47-s + 49-s − 52-s − 0.826·53-s + 1.02·59-s + 63-s + 64-s − 1.58·65-s − 1.12·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(3.515297647\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.515297647\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - p^{3} T \) |
| 13 | \( 1 + p^{3} T \) |
| good | 2 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 3 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 5 | \( 1 - 198 T + p^{6} T^{2} \) |
| 11 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 17 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 19 | \( 1 + 11450 T + p^{6} T^{2} \) |
| 23 | \( 1 + 19710 T + p^{6} T^{2} \) |
| 29 | \( 1 - 47322 T + p^{6} T^{2} \) |
| 31 | \( 1 + 7810 T + p^{6} T^{2} \) |
| 37 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 41 | \( 1 + 124290 T + p^{6} T^{2} \) |
| 43 | \( 1 - 27610 T + p^{6} T^{2} \) |
| 47 | \( 1 + 5346 T + p^{6} T^{2} \) |
| 53 | \( 1 + 123030 T + p^{6} T^{2} \) |
| 59 | \( 1 - 210870 T + p^{6} T^{2} \) |
| 61 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 67 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 71 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 73 | \( 1 + 439234 T + p^{6} T^{2} \) |
| 79 | \( 1 + 953678 T + p^{6} T^{2} \) |
| 83 | \( 1 + 733626 T + p^{6} T^{2} \) |
| 89 | \( 1 - 571230 T + p^{6} T^{2} \) |
| 97 | \( 1 - 1449646 T + p^{6} T^{2} \) |
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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
−12.83647331670688631656737098101, −11.88501177558775461941702360846, −10.39705846678202131649739708827, −10.05627527801126341453083585006, −8.359303879836808243225806356325, −6.99748313039638841020059111967, −6.01895810808712837719263963165, −4.65706435270659407233545758114, −2.30382148576694782669786990818, −1.61896654314278696699824348521,
1.61896654314278696699824348521, 2.30382148576694782669786990818, 4.65706435270659407233545758114, 6.01895810808712837719263963165, 6.99748313039638841020059111967, 8.359303879836808243225806356325, 10.05627527801126341453083585006, 10.39705846678202131649739708827, 11.88501177558775461941702360846, 12.83647331670688631656737098101
