| L(s) = 1 | + 64·4-s − 54·5-s + 610·7-s + 729·9-s − 1.06e3·11-s + 4.09e3·16-s − 9.63e3·17-s − 6.85e3·19-s − 3.45e3·20-s + 2.06e4·23-s − 1.27e4·25-s + 3.90e4·28-s − 3.29e4·35-s + 4.66e4·36-s − 1.42e5·43-s − 6.79e4·44-s − 3.93e4·45-s − 7.51e4·47-s + 2.54e5·49-s + 5.73e4·55-s − 5.70e4·61-s + 4.44e5·63-s + 2.62e5·64-s − 6.16e5·68-s + 3.84e5·73-s − 4.38e5·76-s − 6.47e5·77-s + ⋯ |
| L(s) = 1 | + 4-s − 0.431·5-s + 1.77·7-s + 9-s − 0.797·11-s + 16-s − 1.96·17-s − 19-s − 0.431·20-s + 1.69·23-s − 0.813·25-s + 1.77·28-s − 0.768·35-s + 36-s − 1.79·43-s − 0.797·44-s − 0.431·45-s − 0.723·47-s + 2.16·49-s + 0.344·55-s − 0.251·61-s + 1.77·63-s + 64-s − 1.96·68-s + 0.987·73-s − 76-s − 1.41·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.898477358\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.898477358\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + p^{3} T \) |
| good | 2 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 3 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 5 | \( 1 + 54 T + p^{6} T^{2} \) |
| 7 | \( 1 - 610 T + p^{6} T^{2} \) |
| 11 | \( 1 + 1062 T + p^{6} T^{2} \) |
| 13 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 17 | \( 1 + 9630 T + p^{6} T^{2} \) |
| 23 | \( 1 - 20610 T + p^{6} T^{2} \) |
| 29 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 31 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 37 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 41 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 43 | \( 1 + 142630 T + p^{6} T^{2} \) |
| 47 | \( 1 + 75150 T + p^{6} T^{2} \) |
| 53 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 59 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 61 | \( 1 + 57062 T + p^{6} T^{2} \) |
| 67 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 71 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 73 | \( 1 - 384050 T + p^{6} T^{2} \) |
| 79 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 83 | \( 1 + 1131030 T + p^{6} T^{2} \) |
| 89 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 97 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
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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
−17.19173856724898791512196648239, −15.56603849893962404878872739470, −15.02031263740632618908390578697, −13.08296990250562903187573338817, −11.46022079661992534822829311486, −10.70915727167159081285700550119, −8.277866826031716600387962726016, −7.01719177480492054209852748351, −4.69526103639892601543726183421, −1.91986946735924317516132070768,
1.91986946735924317516132070768, 4.69526103639892601543726183421, 7.01719177480492054209852748351, 8.277866826031716600387962726016, 10.70915727167159081285700550119, 11.46022079661992534822829311486, 13.08296990250562903187573338817, 15.02031263740632618908390578697, 15.56603849893962404878872739470, 17.19173856724898791512196648239
