| L(s) = 1 | + 9·2-s + 71·4-s − 360·5-s + 1.70e3·8-s − 3.24e3·10-s + 4.93e3·11-s − 7.70e3·13-s + 8.58e3·16-s − 2.85e4·17-s + 6.37e4·19-s − 2.55e4·20-s + 4.43e4·22-s − 8.22e4·23-s + 4.74e4·25-s − 6.93e4·26-s + 4.35e5·29-s + 2.92e4·31-s − 5.42e3·32-s − 2.57e5·34-s − 7.09e5·37-s + 5.73e5·38-s − 6.12e5·40-s − 2.50e4·41-s + 4.96e5·43-s + 3.50e5·44-s − 7.40e5·46-s − 1.57e6·47-s + ⋯ |
| L(s) = 1 | + 0.795·2-s + 0.554·4-s − 1.28·5-s + 1.17·8-s − 1.02·10-s + 1.11·11-s − 0.973·13-s + 0.523·16-s − 1.41·17-s + 2.13·19-s − 0.714·20-s + 0.888·22-s − 1.40·23-s + 0.607·25-s − 0.774·26-s + 3.31·29-s + 0.176·31-s − 0.0292·32-s − 1.12·34-s − 2.30·37-s + 1.69·38-s − 1.51·40-s − 0.0567·41-s + 0.951·43-s + 0.619·44-s − 1.12·46-s − 2.21·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.1394556456\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1394556456\) |
| \(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 - 9 T + 5 p T^{2} - 9 p^{7} T^{3} + p^{14} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 72 p T + 3286 p^{2} T^{2} + 72 p^{8} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4932 T + 19797958 T^{2} - 4932 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 7708 T + 118266510 T^{2} + 7708 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 28584 T + 942631150 T^{2} + 28584 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 63728 T + 2569797414 T^{2} - 63728 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 82260 T + 8202169294 T^{2} + 82260 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 435996 T + 80862098782 T^{2} - 435996 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 29240 T + 27798421182 T^{2} - 29240 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 709556 T + 313296440190 T^{2} + 709556 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 25056 T - 26592839954 T^{2} + 25056 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 496216 T + 567160908438 T^{2} - 496216 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 1575000 T + 1564490669086 T^{2} + 1575000 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2057436 T + 3149566808638 T^{2} + 2057436 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 1101024 T + 4521593481622 T^{2} + 1101024 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 28996 T + 5887677435486 T^{2} + 28996 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4480784 T + 16777543143750 T^{2} + 4480784 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 54540 T + 17803030672942 T^{2} + 54540 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 666604 T - 310686963642 T^{2} + 666604 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 2322952 T + 38986658128734 T^{2} - 2322952 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 7384392 T + 61089776413510 T^{2} + 7384392 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 1784448 T + 26626978018894 T^{2} - 1784448 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 16266412 T + 223770781220502 T^{2} + 16266412 p^{7} T^{3} + p^{14} T^{4} \) |
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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
−10.09493961708547879634034894387, −9.968547614333711081972650942301, −9.272997640751395683099154460670, −8.720504216077257882194373392921, −8.294930952929457182289081563441, −7.68465945917795351586671123315, −7.52267434693228877305572806583, −6.72739131448600952137400678386, −6.72077433069965510548521455745, −6.09257585130115295851495324683, −5.08551713988000210369508225297, −4.93266678837707702550657157444, −4.28116631118377142141042968494, −4.20577061650160098586519933028, −3.28351382979333939822765397052, −3.08072195986252936552342989743, −2.27915743442234601128278678385, −1.48124534616895841666433203488, −1.19042906013954488965112173631, −0.06490793773433027755761508615,
0.06490793773433027755761508615, 1.19042906013954488965112173631, 1.48124534616895841666433203488, 2.27915743442234601128278678385, 3.08072195986252936552342989743, 3.28351382979333939822765397052, 4.20577061650160098586519933028, 4.28116631118377142141042968494, 4.93266678837707702550657157444, 5.08551713988000210369508225297, 6.09257585130115295851495324683, 6.72077433069965510548521455745, 6.72739131448600952137400678386, 7.52267434693228877305572806583, 7.68465945917795351586671123315, 8.294930952929457182289081563441, 8.720504216077257882194373392921, 9.272997640751395683099154460670, 9.968547614333711081972650942301, 10.09493961708547879634034894387
