| L(s) = 1 | − 2·7-s − 5·9-s + 11-s + 8·13-s − 4·16-s − 4·17-s − 2·23-s − 9·25-s + 6·37-s − 16·41-s − 3·49-s − 12·53-s + 24·61-s + 10·63-s − 14·67-s − 6·71-s + 8·73-s − 2·77-s + 16·81-s − 12·83-s − 16·91-s − 5·99-s + 4·101-s + 8·112-s + 18·113-s − 40·117-s + 8·119-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 5/3·9-s + 0.301·11-s + 2.21·13-s − 16-s − 0.970·17-s − 0.417·23-s − 9/5·25-s + 0.986·37-s − 2.49·41-s − 3/7·49-s − 1.64·53-s + 3.07·61-s + 1.25·63-s − 1.71·67-s − 0.712·71-s + 0.936·73-s − 0.227·77-s + 16/9·81-s − 1.31·83-s − 1.67·91-s − 0.502·99-s + 0.398·101-s + 0.755·112-s + 1.69·113-s − 3.69·117-s + 0.733·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65219 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65219 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 11 | $C_1$ | \( 1 - T \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.635355093326247349773689827545, −8.982234715263527510913855271701, −8.603539619290756001226038948684, −8.383354580077423849355753624090, −7.73211325750284141433186231976, −6.81130354135159597032281026635, −6.36261389471308870138602900888, −6.10008461377510957946279965770, −5.56513823603813445068893148660, −4.72966200734041708156501051889, −3.84763496937519983566731282131, −3.52931135642754093199166167311, −2.68793343495993615580735780031, −1.77464628596756662135858026846, 0,
1.77464628596756662135858026846, 2.68793343495993615580735780031, 3.52931135642754093199166167311, 3.84763496937519983566731282131, 4.72966200734041708156501051889, 5.56513823603813445068893148660, 6.10008461377510957946279965770, 6.36261389471308870138602900888, 6.81130354135159597032281026635, 7.73211325750284141433186231976, 8.383354580077423849355753624090, 8.603539619290756001226038948684, 8.982234715263527510913855271701, 9.635355093326247349773689827545
