| L(s) = 1 | + 256·4-s − 289·5-s + 527·7-s + 6.56e3·9-s + 2.50e4·11-s + 6.55e4·16-s − 4.24e4·17-s + 1.30e5·19-s − 7.39e4·20-s − 5.34e5·23-s − 3.07e5·25-s + 1.34e5·28-s − 1.52e5·35-s + 1.67e6·36-s + 5.60e6·43-s + 6.40e6·44-s − 1.89e6·45-s − 8.30e6·47-s − 5.48e6·49-s − 7.22e6·55-s − 1.76e7·61-s + 3.45e6·63-s + 1.67e7·64-s − 1.08e7·68-s + 4.38e7·73-s + 3.33e7·76-s + 1.31e7·77-s + ⋯ |
| L(s) = 1 | + 4-s − 0.462·5-s + 0.219·7-s + 9-s + 1.70·11-s + 16-s − 0.508·17-s + 19-s − 0.462·20-s − 1.91·23-s − 0.786·25-s + 0.219·28-s − 0.101·35-s + 36-s + 1.63·43-s + 1.70·44-s − 0.462·45-s − 1.70·47-s − 0.951·49-s − 0.789·55-s − 1.27·61-s + 0.219·63-s + 64-s − 0.508·68-s + 1.54·73-s + 76-s + 0.374·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.138140070\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.138140070\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 - p^{4} T \) |
| good | 2 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 3 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 5 | \( 1 + 289 T + p^{8} T^{2} \) |
| 7 | \( 1 - 527 T + p^{8} T^{2} \) |
| 11 | \( 1 - 25007 T + p^{8} T^{2} \) |
| 13 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 17 | \( 1 + 42433 T + p^{8} T^{2} \) |
| 23 | \( 1 + 534718 T + p^{8} T^{2} \) |
| 29 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 31 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 37 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 41 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 43 | \( 1 - 5602127 T + p^{8} T^{2} \) |
| 47 | \( 1 + 8302513 T + p^{8} T^{2} \) |
| 53 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 59 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 61 | \( 1 + 17661793 T + p^{8} T^{2} \) |
| 67 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 71 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 73 | \( 1 - 43864607 T + p^{8} T^{2} \) |
| 79 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 83 | \( 1 + 62676958 T + p^{8} T^{2} \) |
| 89 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 97 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
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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
−16.36952451055305255165590397125, −15.51622042883369291165004124031, −14.17425446297332544648741371126, −12.24934798103107471074673008704, −11.38320420279666586417593563349, −9.731655870863968601620853834297, −7.70918237628389585928084946933, −6.40453771600558010959518781301, −3.93767592201659984567242924152, −1.57987607059496258311442911275,
1.57987607059496258311442911275, 3.93767592201659984567242924152, 6.40453771600558010959518781301, 7.70918237628389585928084946933, 9.731655870863968601620853834297, 11.38320420279666586417593563349, 12.24934798103107471074673008704, 14.17425446297332544648741371126, 15.51622042883369291165004124031, 16.36952451055305255165590397125
