| L(s) = 1 | + 2·7-s + 18·19-s + 4·25-s − 6·31-s + 10·37-s − 44·43-s − 11·49-s − 12·61-s + 14·67-s + 54·73-s − 22·79-s + 30·103-s + 10·109-s − 4·121-s + 127-s + 131-s + 36·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 46·169-s + 173-s + 8·175-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 4.12·19-s + 4/5·25-s − 1.07·31-s + 1.64·37-s − 6.70·43-s − 1.57·49-s − 1.53·61-s + 1.71·67-s + 6.32·73-s − 2.47·79-s + 2.95·103-s + 0.957·109-s − 0.363·121-s + 0.0887·127-s + 0.0873·131-s + 3.12·133-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 + 0.0773·167-s + 3.53·169-s + 0.0760·173-s + 0.604·175-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.162717438\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.162717438\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| good | 5 | $C_2^3$ | \( 1 - 4 T^{2} - 9 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^3$ | \( 1 + 4 T^{2} - 105 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 10 T^{2} - 189 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 - T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 5 T - 12 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 68 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{4} \) |
| 47 | $C_2^3$ | \( 1 - 88 T^{2} + 5535 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 34 T^{2} - 1653 T^{4} + 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^3$ | \( 1 - 94 T^{2} + 5355 T^{4} - 94 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 6 T + 73 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2}( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 11 T + 42 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 + 38 T^{2} - 6477 T^{4} + 38 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 - 182 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−8.826132711472494254159353082158, −8.436413443175717186009511848225, −8.090514353022859697083395214164, −8.050154204324423828593500120443, −7.84766420295975051561484822185, −7.62108095900743968520313492496, −7.08696045026379803266787478060, −6.91630955251243010655675039666, −6.85266765662240481903303805564, −6.43287194056498319643870936649, −6.16334873165294198695912717051, −5.65361615539146904128759010102, −5.36974936038712407766431862445, −5.19425963867059200122406711247, −4.98688067689919651731722617299, −4.68206038359731570134866659811, −4.57474437931164353483963472192, −3.56719629838045156224940560503, −3.53388503834659934706710823264, −3.25574812476184287349372974983, −3.16853055523483917202613613921, −2.37366943079236095456801294297, −1.81632771225251732995576875751, −1.48893953575493608715093514627, −0.858704779014231631331359456110,
0.858704779014231631331359456110, 1.48893953575493608715093514627, 1.81632771225251732995576875751, 2.37366943079236095456801294297, 3.16853055523483917202613613921, 3.25574812476184287349372974983, 3.53388503834659934706710823264, 3.56719629838045156224940560503, 4.57474437931164353483963472192, 4.68206038359731570134866659811, 4.98688067689919651731722617299, 5.19425963867059200122406711247, 5.36974936038712407766431862445, 5.65361615539146904128759010102, 6.16334873165294198695912717051, 6.43287194056498319643870936649, 6.85266765662240481903303805564, 6.91630955251243010655675039666, 7.08696045026379803266787478060, 7.62108095900743968520313492496, 7.84766420295975051561484822185, 8.050154204324423828593500120443, 8.090514353022859697083395214164, 8.436413443175717186009511848225, 8.826132711472494254159353082158
