L(s) = 1 | + 2·7-s − 9-s − 4·11-s + 2·25-s − 14·29-s + 10·37-s + 8·43-s − 3·49-s + 2·53-s − 2·63-s − 4·67-s − 8·77-s + 4·79-s + 81-s + 4·99-s − 16·107-s + 2·109-s + 28·113-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
L(s) = 1 | + 0.755·7-s − 1/3·9-s − 1.20·11-s + 2/5·25-s − 2.59·29-s + 1.64·37-s + 1.21·43-s − 3/7·49-s + 0.274·53-s − 0.251·63-s − 0.488·67-s − 0.911·77-s + 0.450·79-s + 1/9·81-s + 0.402·99-s − 1.54·107-s + 0.191·109-s + 2.63·113-s − 0.545·121-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 + 0.0783·163-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 | $C_2$ | \( 1 + T^{2} \) |
| 7 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
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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
−7.78245137412435038045125827543, −7.64135024189773811621954601746, −7.32749690348162622935331198129, −6.60097882848986377599372816654, −6.05638149647234243259577550038, −5.68532271651097360611006677500, −5.22167597519433024824500885074, −4.88599899621971466303304395158, −4.21573580818128666730573804230, −3.80690805477040688503515855992, −3.07773167527900354467457746455, −2.49754261012562220489332233149, −2.03475856903185336743352365942, −1.15429076906364918568765649279, 0,
1.15429076906364918568765649279, 2.03475856903185336743352365942, 2.49754261012562220489332233149, 3.07773167527900354467457746455, 3.80690805477040688503515855992, 4.21573580818128666730573804230, 4.88599899621971466303304395158, 5.22167597519433024824500885074, 5.68532271651097360611006677500, 6.05638149647234243259577550038, 6.60097882848986377599372816654, 7.32749690348162622935331198129, 7.64135024189773811621954601746, 7.78245137412435038045125827543