| L(s) = 1 | + 2·7-s + 9-s + 2·11-s − 4·23-s − 6·25-s − 4·29-s − 6·43-s − 3·49-s − 12·53-s + 2·63-s + 6·67-s − 4·71-s + 4·77-s + 81-s + 2·99-s − 6·107-s − 12·109-s − 20·113-s − 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 8·161-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 1/3·9-s + 0.603·11-s − 0.834·23-s − 6/5·25-s − 0.742·29-s − 0.914·43-s − 3/7·49-s − 1.64·53-s + 0.251·63-s + 0.733·67-s − 0.474·71-s + 0.455·77-s + 1/9·81-s + 0.201·99-s − 0.580·107-s − 1.14·109-s − 1.88·113-s − 1.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s − 0.630·161-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 903168 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 903168 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 126 T^{2} + p^{2} T^{4} \) |
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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
−7.944027950251713854382484468373, −7.65816326162839818071076535162, −7.11368432812811231417077892456, −6.54481984333820376656547165448, −6.27810715795333523731959907136, −5.69249307324895052836728130463, −5.20619731944266437912850129345, −4.78557381840291381873988399483, −4.12185761127986315414716401036, −3.87724421829152808119356849849, −3.25349697730398558304120822787, −2.45622839663656188665656384206, −1.76583582508951215211160323468, −1.38137927096995190084839050054, 0,
1.38137927096995190084839050054, 1.76583582508951215211160323468, 2.45622839663656188665656384206, 3.25349697730398558304120822787, 3.87724421829152808119356849849, 4.12185761127986315414716401036, 4.78557381840291381873988399483, 5.20619731944266437912850129345, 5.69249307324895052836728130463, 6.27810715795333523731959907136, 6.54481984333820376656547165448, 7.11368432812811231417077892456, 7.65816326162839818071076535162, 7.944027950251713854382484468373
