| L(s) = 1 | − 3-s + 9-s + 8·23-s − 10·25-s − 27-s + 8·29-s + 16·43-s − 24·47-s − 10·49-s − 24·53-s + 8·67-s − 8·69-s − 8·71-s − 20·73-s + 10·75-s + 81-s − 8·87-s − 12·97-s + 8·101-s − 6·121-s + 127-s − 16·129-s + 131-s + 137-s + 139-s + 24·141-s + 10·147-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 1.66·23-s − 2·25-s − 0.192·27-s + 1.48·29-s + 2.43·43-s − 3.50·47-s − 1.42·49-s − 3.29·53-s + 0.977·67-s − 0.963·69-s − 0.949·71-s − 2.34·73-s + 1.15·75-s + 1/9·81-s − 0.857·87-s − 1.21·97-s + 0.796·101-s − 0.545·121-s + 0.0887·127-s − 1.40·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.02·141-s + 0.824·147-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | | \( 1 \) |
| 3 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−8.272861522300808832842531458007, −7.83756825226352882885863464519, −7.56237844869816331895572212869, −6.86617713649335228477722027286, −6.40139715881748782453956361829, −6.19174829158795733764599049329, −5.52928834758807112839354938677, −4.99207876302239290655432150674, −4.57667813854686016923499181308, −4.14300745369301184814326433789, −3.12662782741322079682173901720, −3.04605515966584879794132221618, −1.89290593869789230240007927288, −1.27109672072212401682622254955, 0,
1.27109672072212401682622254955, 1.89290593869789230240007927288, 3.04605515966584879794132221618, 3.12662782741322079682173901720, 4.14300745369301184814326433789, 4.57667813854686016923499181308, 4.99207876302239290655432150674, 5.52928834758807112839354938677, 6.19174829158795733764599049329, 6.40139715881748782453956361829, 6.86617713649335228477722027286, 7.56237844869816331895572212869, 7.83756825226352882885863464519, 8.272861522300808832842531458007
