| L(s) = 1 | − 2·4-s + 4·16-s − 8·19-s − 12·23-s − 12·29-s − 16·43-s − 4·49-s + 24·53-s − 8·64-s − 16·67-s − 28·73-s + 16·76-s + 24·92-s + 20·97-s − 12·101-s + 24·116-s + 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + ⋯ |
| L(s) = 1 | − 4-s + 16-s − 1.83·19-s − 2.50·23-s − 2.22·29-s − 2.43·43-s − 4/7·49-s + 3.29·53-s − 64-s − 1.95·67-s − 3.27·73-s + 1.83·76-s + 2.50·92-s + 2.03·97-s − 1.19·101-s + 2.22·116-s + 1.81·121-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 + 0.0773·167-s + 8/13·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 56 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 116 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 128 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−8.943988129895287139199922873902, −8.624068777247899082972725224282, −8.454115931298661236787194428559, −8.049317074485377000421313087423, −7.37236661811206806249157074321, −7.36333165492618015195395587155, −6.66949795278718274496167074821, −6.02980566535409507150749377977, −5.88354853880028814942318557493, −5.58709075838795321300631670447, −4.85393781871152502645944427568, −4.47406466746149651242449255483, −4.11181320787517250526126502702, −3.67021397352078189010026446535, −3.34560043828901261758983893304, −2.38911572744647491431735255709, −1.97639686754233461993743636092, −1.42092304265090400906119991156, 0, 0,
1.42092304265090400906119991156, 1.97639686754233461993743636092, 2.38911572744647491431735255709, 3.34560043828901261758983893304, 3.67021397352078189010026446535, 4.11181320787517250526126502702, 4.47406466746149651242449255483, 4.85393781871152502645944427568, 5.58709075838795321300631670447, 5.88354853880028814942318557493, 6.02980566535409507150749377977, 6.66949795278718274496167074821, 7.36333165492618015195395587155, 7.37236661811206806249157074321, 8.049317074485377000421313087423, 8.454115931298661236787194428559, 8.624068777247899082972725224282, 8.943988129895287139199922873902
