| L(s) = 1 | − 3·2-s + 8·3-s + 4-s − 6·5-s − 24·6-s + 28·7-s + 21·8-s + 37·9-s + 18·10-s + 24·11-s + 8·12-s − 58·13-s − 84·14-s − 48·15-s − 71·16-s − 111·18-s + 116·19-s − 6·20-s + 224·21-s − 72·22-s + 60·23-s + 168·24-s − 89·25-s + 174·26-s + 80·27-s + 28·28-s − 30·29-s + ⋯ |
| L(s) = 1 | − 1.06·2-s + 1.53·3-s + 1/8·4-s − 0.536·5-s − 1.63·6-s + 1.51·7-s + 0.928·8-s + 1.37·9-s + 0.569·10-s + 0.657·11-s + 0.192·12-s − 1.23·13-s − 1.60·14-s − 0.826·15-s − 1.10·16-s − 1.45·18-s + 1.40·19-s − 0.0670·20-s + 2.32·21-s − 0.697·22-s + 0.543·23-s + 1.42·24-s − 0.711·25-s + 1.31·26-s + 0.570·27-s + 0.188·28-s − 0.192·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.859652352\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.859652352\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 + 3 T + p^{3} T^{2} \) |
| 3 | \( 1 - 8 T + p^{3} T^{2} \) |
| 5 | \( 1 + 6 T + p^{3} T^{2} \) |
| 7 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 11 | \( 1 - 24 T + p^{3} T^{2} \) |
| 13 | \( 1 + 58 T + p^{3} T^{2} \) |
| 19 | \( 1 - 116 T + p^{3} T^{2} \) |
| 23 | \( 1 - 60 T + p^{3} T^{2} \) |
| 29 | \( 1 + 30 T + p^{3} T^{2} \) |
| 31 | \( 1 - 172 T + p^{3} T^{2} \) |
| 37 | \( 1 - 58 T + p^{3} T^{2} \) |
| 41 | \( 1 - 342 T + p^{3} T^{2} \) |
| 43 | \( 1 + 148 T + p^{3} T^{2} \) |
| 47 | \( 1 - 288 T + p^{3} T^{2} \) |
| 53 | \( 1 - 6 p T + p^{3} T^{2} \) |
| 59 | \( 1 - 252 T + p^{3} T^{2} \) |
| 61 | \( 1 + 110 T + p^{3} T^{2} \) |
| 67 | \( 1 + 484 T + p^{3} T^{2} \) |
| 71 | \( 1 - 708 T + p^{3} T^{2} \) |
| 73 | \( 1 + 362 T + p^{3} T^{2} \) |
| 79 | \( 1 - 484 T + p^{3} T^{2} \) |
| 83 | \( 1 - 756 T + p^{3} T^{2} \) |
| 89 | \( 1 + 774 T + p^{3} T^{2} \) |
| 97 | \( 1 - 382 T + p^{3} T^{2} \) |
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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
−11.24547591967824033871491306035, −9.987133135915180399369928210635, −9.271818194518263795843583613821, −8.509797947694011142831070392484, −7.67526847741221852747540932496, −7.40647485149367352777901705693, −4.95497563628886832380705400934, −3.97895843677252400172527551144, −2.39742096361137259946219217649, −1.16122397312693974776619485510,
1.16122397312693974776619485510, 2.39742096361137259946219217649, 3.97895843677252400172527551144, 4.95497563628886832380705400934, 7.40647485149367352777901705693, 7.67526847741221852747540932496, 8.509797947694011142831070392484, 9.271818194518263795843583613821, 9.987133135915180399369928210635, 11.24547591967824033871491306035
