L(s) = 1 | − 3-s + 3·4-s + 9-s − 3·12-s + 5·16-s − 25-s − 27-s + 3·36-s − 10·37-s − 5·48-s − 14·49-s + 3·64-s − 24·67-s − 20·73-s + 75-s + 81-s − 3·100-s − 3·108-s + 10·111-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 5·144-s + 14·147-s − 30·148-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 3/2·4-s + 1/3·9-s − 0.866·12-s + 5/4·16-s − 1/5·25-s − 0.192·27-s + 1/2·36-s − 1.64·37-s − 0.721·48-s − 2·49-s + 3/8·64-s − 2.93·67-s − 2.34·73-s + 0.115·75-s + 1/9·81-s − 0.299·100-s − 0.288·108-s + 0.949·111-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5/12·144-s + 1.15·147-s − 2.46·148-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 924075 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 924075 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$ | \( 1 + T \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 37 | $C_2$ | \( 1 + 10 T + p T^{2} \) |
good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−7.67799447246984285256141034859, −7.45907484152201151608049185099, −7.07321471777940170771721600626, −6.52540889732913632888215536500, −6.26737125696246899852677234655, −5.85903094197330009063002786748, −5.31376667511682693604238036981, −4.84325595526585691365215254100, −4.28051417709748155287284391520, −3.59913768228706162043949303689, −3.04702739671139785549038649232, −2.60046267825816156584494055207, −1.68367772004210019912876523382, −1.48466102625364914227122770826, 0,
1.48466102625364914227122770826, 1.68367772004210019912876523382, 2.60046267825816156584494055207, 3.04702739671139785549038649232, 3.59913768228706162043949303689, 4.28051417709748155287284391520, 4.84325595526585691365215254100, 5.31376667511682693604238036981, 5.85903094197330009063002786748, 6.26737125696246899852677234655, 6.52540889732913632888215536500, 7.07321471777940170771721600626, 7.45907484152201151608049185099, 7.67799447246984285256141034859