| L(s) = 1 | + 2·3-s + 9-s + 3·17-s − 13·19-s + 8·25-s − 4·27-s − 3·41-s + 11·43-s − 4·49-s + 6·51-s − 26·57-s + 6·59-s − 7·67-s + 4·73-s + 16·75-s − 11·81-s + 3·83-s − 15·89-s + 16·97-s − 18·107-s − 13·121-s − 6·123-s + 127-s + 22·129-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/3·9-s + 0.727·17-s − 2.98·19-s + 8/5·25-s − 0.769·27-s − 0.468·41-s + 1.67·43-s − 4/7·49-s + 0.840·51-s − 3.44·57-s + 0.781·59-s − 0.855·67-s + 0.468·73-s + 1.84·75-s − 1.22·81-s + 0.329·83-s − 1.58·89-s + 1.62·97-s − 1.74·107-s − 1.18·121-s − 0.541·123-s + 0.0887·127-s + 1.93·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2663424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2663424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 17 | $C_2$ | \( 1 - 3 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + T + p T^{2} ) \) |
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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
−7.67697175885132419629651347393, −6.98401244346476020498985387261, −6.51309990352212175843481360236, −6.37635423755938835838879348279, −5.70356457551594138995171392474, −5.29785597292372860296269462665, −4.68418763686885835724844742849, −4.24656304778275260585707031203, −3.90010753658206341209329021642, −3.35958859696658407343370145347, −2.74187028668414837680964192487, −2.45209025666650315489625454989, −1.89285042363417255815900435727, −1.13701063538836866812130735895, 0,
1.13701063538836866812130735895, 1.89285042363417255815900435727, 2.45209025666650315489625454989, 2.74187028668414837680964192487, 3.35958859696658407343370145347, 3.90010753658206341209329021642, 4.24656304778275260585707031203, 4.68418763686885835724844742849, 5.29785597292372860296269462665, 5.70356457551594138995171392474, 6.37635423755938835838879348279, 6.51309990352212175843481360236, 6.98401244346476020498985387261, 7.67697175885132419629651347393
