| L(s) = 1 | − 2·4-s + 8·7-s + 6·9-s + 3·16-s − 20·25-s − 16·28-s − 12·36-s − 8·37-s − 8·43-s + 34·49-s + 48·63-s − 4·64-s − 16·67-s + 32·79-s + 27·81-s + 40·100-s − 80·109-s + 24·112-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 18·144-s + 16·148-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 4-s + 3.02·7-s + 2·9-s + 3/4·16-s − 4·25-s − 3.02·28-s − 2·36-s − 1.31·37-s − 1.21·43-s + 34/7·49-s + 6.04·63-s − 1/2·64-s − 1.95·67-s + 3.60·79-s + 3·81-s + 4·100-s − 7.66·109-s + 2.26·112-s − 0.181·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 3/2·144-s + 1.31·148-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.765566032\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.765566032\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 - 106 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 146 T^{2} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.987607169942069840158269168813, −7.81279471412227562108202160655, −7.61121117835497331651037045802, −7.58303265437170784155467966210, −6.94540834172087397037375155189, −6.86438870366687228934610490361, −6.71938939171693366053646255990, −6.18911421087111104326905932366, −5.85855308463100640542285659787, −5.64691619956599129925532694512, −5.38422497104102263906327643677, −5.07782702454631711925128064231, −4.93325801611994502481187269447, −4.69999864798053547169839904242, −4.37275262194764698851335905721, −4.04976401376089661775401427431, −3.91331429197674510898519512920, −3.84414947246844531381387588615, −3.33237280938223814320612671187, −2.71508890791417022891425312850, −2.17769200074351655647976243777, −1.72227542664204167521647047351, −1.61092397060891597455556310135, −1.58643496396095380548099275407, −0.59658248094432583186866255363,
0.59658248094432583186866255363, 1.58643496396095380548099275407, 1.61092397060891597455556310135, 1.72227542664204167521647047351, 2.17769200074351655647976243777, 2.71508890791417022891425312850, 3.33237280938223814320612671187, 3.84414947246844531381387588615, 3.91331429197674510898519512920, 4.04976401376089661775401427431, 4.37275262194764698851335905721, 4.69999864798053547169839904242, 4.93325801611994502481187269447, 5.07782702454631711925128064231, 5.38422497104102263906327643677, 5.64691619956599129925532694512, 5.85855308463100640542285659787, 6.18911421087111104326905932366, 6.71938939171693366053646255990, 6.86438870366687228934610490361, 6.94540834172087397037375155189, 7.58303265437170784155467966210, 7.61121117835497331651037045802, 7.81279471412227562108202160655, 7.987607169942069840158269168813
