| L(s) = 1 | + 3-s + 9-s − 4·13-s − 8·23-s − 6·25-s + 27-s − 20·37-s − 4·39-s − 16·47-s + 49-s + 8·59-s − 20·61-s − 8·69-s + 8·71-s + 4·73-s − 6·75-s + 81-s + 8·83-s + 20·97-s + 32·107-s + 28·109-s − 20·111-s − 4·117-s − 22·121-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 1.10·13-s − 1.66·23-s − 6/5·25-s + 0.192·27-s − 3.28·37-s − 0.640·39-s − 2.33·47-s + 1/7·49-s + 1.04·59-s − 2.56·61-s − 0.963·69-s + 0.949·71-s + 0.468·73-s − 0.692·75-s + 1/9·81-s + 0.878·83-s + 2.03·97-s + 3.09·107-s + 2.68·109-s − 1.89·111-s − 0.369·117-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169344 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169344 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−8.820314983809012661425051705328, −8.614680372438593469438231251008, −7.86152616392232705724947299443, −7.65994621019403063529610936786, −7.17945773905277725475919160716, −6.42136658293515809263418102895, −6.19452637109873977406389525575, −5.25019213638990204722565722077, −4.99862586128493531522947252735, −4.26888101159701758252770076566, −3.56921408081238054859871707771, −3.21569388283931405752209731146, −2.06515692945354055956624655608, −1.87750590194886957605795356311, 0,
1.87750590194886957605795356311, 2.06515692945354055956624655608, 3.21569388283931405752209731146, 3.56921408081238054859871707771, 4.26888101159701758252770076566, 4.99862586128493531522947252735, 5.25019213638990204722565722077, 6.19452637109873977406389525575, 6.42136658293515809263418102895, 7.17945773905277725475919160716, 7.65994621019403063529610936786, 7.86152616392232705724947299443, 8.614680372438593469438231251008, 8.820314983809012661425051705328
