| L(s) = 1 | + 3-s + 4-s + 4·7-s + 9-s + 12-s + 12·13-s + 16-s + 2·19-s + 4·21-s + 25-s + 27-s + 4·28-s − 4·31-s + 36-s − 4·37-s + 12·39-s − 16·43-s + 48-s − 2·49-s + 12·52-s + 2·57-s + 4·61-s + 4·63-s + 64-s + 8·67-s − 20·73-s + 75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/2·4-s + 1.51·7-s + 1/3·9-s + 0.288·12-s + 3.32·13-s + 1/4·16-s + 0.458·19-s + 0.872·21-s + 1/5·25-s + 0.192·27-s + 0.755·28-s − 0.718·31-s + 1/6·36-s − 0.657·37-s + 1.92·39-s − 2.43·43-s + 0.144·48-s − 2/7·49-s + 1.66·52-s + 0.264·57-s + 0.512·61-s + 0.503·63-s + 1/8·64-s + 0.977·67-s − 2.34·73-s + 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 974700 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 974700 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.664148048\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.664148048\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−8.044107996050708391409834608218, −7.948201996751845026195458906853, −7.37575424356828643155557263420, −6.67717098920958314152394624363, −6.51914956387136270852546336776, −5.95339000657840846209039457086, −5.35708181227167532787761077019, −5.11687773951392317925695470662, −4.30032808357235128666275695206, −3.93643728930441993139518178511, −3.29298652740753695701377317054, −3.11063806056870265452985737851, −1.89790938700513335557070325628, −1.63018968978531456834372202160, −1.12737143444698159586204498120,
1.12737143444698159586204498120, 1.63018968978531456834372202160, 1.89790938700513335557070325628, 3.11063806056870265452985737851, 3.29298652740753695701377317054, 3.93643728930441993139518178511, 4.30032808357235128666275695206, 5.11687773951392317925695470662, 5.35708181227167532787761077019, 5.95339000657840846209039457086, 6.51914956387136270852546336776, 6.67717098920958314152394624363, 7.37575424356828643155557263420, 7.948201996751845026195458906853, 8.044107996050708391409834608218
