| L(s) = 1 | + 2·2-s + 4-s − 2·8-s − 4·11-s − 4·16-s − 8·22-s − 8·23-s + 2·25-s − 8·29-s − 2·32-s − 20·37-s + 8·43-s − 4·44-s − 16·46-s + 4·50-s − 4·53-s − 16·58-s + 3·64-s − 24·67-s + 48·71-s − 40·74-s + 8·79-s + 16·86-s + 8·88-s − 8·92-s + 2·100-s − 8·106-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1/2·4-s − 0.707·8-s − 1.20·11-s − 16-s − 1.70·22-s − 1.66·23-s + 2/5·25-s − 1.48·29-s − 0.353·32-s − 3.28·37-s + 1.21·43-s − 0.603·44-s − 2.35·46-s + 0.565·50-s − 0.549·53-s − 2.10·58-s + 3/8·64-s − 2.93·67-s + 5.69·71-s − 4.64·74-s + 0.900·79-s + 1.72·86-s + 0.852·88-s − 0.834·92-s + 1/5·100-s − 0.777·106-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.798811441\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.798811441\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) |
| 11 | $C_2^2$ | \( ( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 17 | $C_2^3$ | \( 1 - 32 T^{2} + 735 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^3$ | \( 1 + 12 T^{2} - 217 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 31 | $C_2^3$ | \( 1 + 10 T^{2} - 861 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 16 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 47 | $C_2^3$ | \( 1 - 86 T^{2} + 5187 T^{4} - 86 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + 2 T - 49 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 116 T^{2} + 9975 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^3$ | \( 1 - 114 T^{2} + 9275 T^{4} - 114 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2^2$ | \( ( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 - 144 T^{2} + 15407 T^{4} - 144 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 68 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - 128 T^{2} + 8463 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 + 96 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−7.27611633781654403058560036113, −7.05924573167588351249902334772, −6.79053802981871482084737415493, −6.54419972848753534803873742700, −6.07583927514816803550481876316, −6.02625984748882278786600641477, −5.97372687845616218937895171056, −5.48946469700608741557806251110, −5.35845812057103295600371167702, −5.29064751709623561432405187482, −4.90423397285373384879936245087, −4.64576506204875424110696668340, −4.52795059817113101429025942372, −4.19386852039087308500281209603, −3.92203131165310775730625886618, −3.63029029345049699346774820208, −3.27613870875071603727182958902, −3.19002201008767535183516570283, −3.16159231578217448431540122192, −2.37615008901555407300485293327, −2.07450558937990508322260981923, −2.00881355078638821036491364042, −1.75523630226212979574070154714, −0.77223306029558540932517534850, −0.38739379190127039408092108108,
0.38739379190127039408092108108, 0.77223306029558540932517534850, 1.75523630226212979574070154714, 2.00881355078638821036491364042, 2.07450558937990508322260981923, 2.37615008901555407300485293327, 3.16159231578217448431540122192, 3.19002201008767535183516570283, 3.27613870875071603727182958902, 3.63029029345049699346774820208, 3.92203131165310775730625886618, 4.19386852039087308500281209603, 4.52795059817113101429025942372, 4.64576506204875424110696668340, 4.90423397285373384879936245087, 5.29064751709623561432405187482, 5.35845812057103295600371167702, 5.48946469700608741557806251110, 5.97372687845616218937895171056, 6.02625984748882278786600641477, 6.07583927514816803550481876316, 6.54419972848753534803873742700, 6.79053802981871482084737415493, 7.05924573167588351249902334772, 7.27611633781654403058560036113
