| L(s) = 1 | − 7-s − 4·9-s + 6·11-s − 2·23-s − 2·25-s + 4·29-s + 8·37-s − 10·43-s + 49-s − 2·53-s + 4·63-s + 8·67-s − 16·71-s − 6·77-s − 4·79-s + 7·81-s − 24·99-s − 12·107-s − 20·109-s − 16·113-s + 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 4/3·9-s + 1.80·11-s − 0.417·23-s − 2/5·25-s + 0.742·29-s + 1.31·37-s − 1.52·43-s + 1/7·49-s − 0.274·53-s + 0.503·63-s + 0.977·67-s − 1.89·71-s − 0.683·77-s − 0.450·79-s + 7/9·81-s − 2.41·99-s − 1.16·107-s − 1.91·109-s − 1.50·113-s + 1.27·121-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1404928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1404928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_1$ | \( 1 + T \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 68 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
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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
−7.906062403938228542679312398505, −7.21266210692081217760995062289, −6.72266199817642079436113278419, −6.34042616063462477383315538038, −6.19301546835800892347944132928, −5.52097689825881924367323162295, −5.22610854033153588338685358423, −4.41018481923210917656030901512, −4.12314550792245600494335570182, −3.57963294693398734387319060901, −3.01206551410525039614809972986, −2.60422443330861622954426319001, −1.76501277401255988744889364793, −1.09057764479192131940372108563, 0,
1.09057764479192131940372108563, 1.76501277401255988744889364793, 2.60422443330861622954426319001, 3.01206551410525039614809972986, 3.57963294693398734387319060901, 4.12314550792245600494335570182, 4.41018481923210917656030901512, 5.22610854033153588338685358423, 5.52097689825881924367323162295, 6.19301546835800892347944132928, 6.34042616063462477383315538038, 6.72266199817642079436113278419, 7.21266210692081217760995062289, 7.906062403938228542679312398505
