| L(s) = 1 | + 2·2-s + 3-s + 2·4-s − 4·5-s + 2·6-s − 7-s − 8·10-s + 2·12-s + 2·13-s − 2·14-s − 4·15-s − 4·17-s + 3·19-s − 8·20-s − 21-s + 8·23-s + 5·25-s + 4·26-s − 2·28-s − 6·29-s − 8·30-s + 5·31-s + 8·32-s − 8·34-s + 4·35-s − 3·37-s + 6·38-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s − 1.78·5-s + 0.816·6-s − 0.377·7-s − 2.52·10-s + 0.577·12-s + 0.554·13-s − 0.534·14-s − 1.03·15-s − 0.970·17-s + 0.688·19-s − 1.78·20-s − 0.218·21-s + 1.66·23-s + 25-s + 0.784·26-s − 0.377·28-s − 1.11·29-s − 1.46·30-s + 0.898·31-s + 1.41·32-s − 1.37·34-s + 0.676·35-s − 0.493·37-s + 0.973·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.202100703\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.202100703\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_4\times C_2$ | \( 1 - p T + p T^{2} - p^{2} T^{4} + p^{3} T^{6} - p^{4} T^{7} + p^{4} T^{8} \) |
| 5 | $C_4\times C_2$ | \( 1 + 4 T + 11 T^{2} + 24 T^{3} + 41 T^{4} + 24 p T^{5} + 11 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_4\times C_2$ | \( 1 + T - 6 T^{2} - 13 T^{3} + 29 T^{4} - 13 p T^{5} - 6 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_4\times C_2$ | \( 1 - 2 T - 9 T^{2} + 44 T^{3} + 29 T^{4} + 44 p T^{5} - 9 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_4\times C_2$ | \( 1 + 4 T - T^{2} - 72 T^{3} - 271 T^{4} - 72 p T^{5} - p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_4\times C_2$ | \( 1 - 3 T - 10 T^{2} + 87 T^{3} - 71 T^{4} + 87 p T^{5} - 10 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 29 | $C_4\times C_2$ | \( 1 + 6 T + 7 T^{2} - 132 T^{3} - 995 T^{4} - 132 p T^{5} + 7 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_4\times C_2$ | \( 1 - 5 T - 6 T^{2} + 185 T^{3} - 739 T^{4} + 185 p T^{5} - 6 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_4\times C_2$ | \( 1 + 3 T - 28 T^{2} - 195 T^{3} + 451 T^{4} - 195 p T^{5} - 28 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_4\times C_2$ | \( 1 - 2 T - 37 T^{2} + 156 T^{3} + 1205 T^{4} + 156 p T^{5} - 37 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
| 47 | $C_4\times C_2$ | \( 1 + 2 T - 43 T^{2} - 180 T^{3} + 1661 T^{4} - 180 p T^{5} - 43 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_4\times C_2$ | \( 1 + 6 T - 17 T^{2} - 420 T^{3} - 1619 T^{4} - 420 p T^{5} - 17 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_4\times C_2$ | \( 1 - 10 T + 41 T^{2} + 180 T^{3} - 4219 T^{4} + 180 p T^{5} + 41 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_4\times C_2$ | \( 1 + 3 T - 52 T^{2} - 339 T^{3} + 2155 T^{4} - 339 p T^{5} - 52 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 71 | $C_4\times C_2$ | \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 73 | $C_4\times C_2$ | \( 1 - 11 T + 48 T^{2} + 275 T^{3} - 6529 T^{4} + 275 p T^{5} + 48 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_4\times C_2$ | \( 1 + 11 T + 42 T^{2} - 407 T^{3} - 7795 T^{4} - 407 p T^{5} + 42 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_4\times C_2$ | \( 1 + 6 T - 47 T^{2} - 780 T^{3} - 779 T^{4} - 780 p T^{5} - 47 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
| 97 | $C_4\times C_2$ | \( 1 + 5 T - 72 T^{2} - 845 T^{3} + 2759 T^{4} - 845 p T^{5} - 72 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−8.128884021528328490221600093320, −7.79657745460773506535140235869, −7.71412500872165265114027978382, −7.70442642977038168650644535156, −7.30393750512752986916598454666, −7.09189010786535036187313322949, −6.81181152174878746713040403945, −6.47256653652740055567355236796, −6.12776906322719890434001836234, −5.81396678845320349125158428825, −5.80337222659154988070855351734, −5.58438627778757886434624205485, −4.83959908000021541539292423472, −4.77743798696406079333418042085, −4.59065229239504166699727938150, −4.14403782812745651104495231310, −3.97961642880742711257925137030, −3.58678988631292486509771966623, −3.56149018517137076201341915309, −3.12504292312340698071910735657, −2.72506616924827748787200966047, −2.36676814935249859874227146197, −2.28608464492803981083177642486, −0.938142977752789380568467966929, −0.872597404376471983115521268714,
0.872597404376471983115521268714, 0.938142977752789380568467966929, 2.28608464492803981083177642486, 2.36676814935249859874227146197, 2.72506616924827748787200966047, 3.12504292312340698071910735657, 3.56149018517137076201341915309, 3.58678988631292486509771966623, 3.97961642880742711257925137030, 4.14403782812745651104495231310, 4.59065229239504166699727938150, 4.77743798696406079333418042085, 4.83959908000021541539292423472, 5.58438627778757886434624205485, 5.80337222659154988070855351734, 5.81396678845320349125158428825, 6.12776906322719890434001836234, 6.47256653652740055567355236796, 6.81181152174878746713040403945, 7.09189010786535036187313322949, 7.30393750512752986916598454666, 7.70442642977038168650644535156, 7.71412500872165265114027978382, 7.79657745460773506535140235869, 8.128884021528328490221600093320
