| L(s) = 1 | + 3·3-s − 5·5-s − 4·7-s + 2·9-s − 8·13-s − 15·15-s + 4·17-s − 12·21-s − 6·23-s + 4·25-s − 4·27-s − 8·29-s + 7·31-s + 20·35-s − 8·37-s − 24·39-s + 15·41-s − 9·43-s − 10·45-s − 6·47-s + 10·49-s + 12·51-s − 17·53-s − 5·61-s − 8·63-s + 40·65-s − 9·67-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 2.23·5-s − 1.51·7-s + 2/3·9-s − 2.21·13-s − 3.87·15-s + 0.970·17-s − 2.61·21-s − 1.25·23-s + 4/5·25-s − 0.769·27-s − 1.48·29-s + 1.25·31-s + 3.38·35-s − 1.31·37-s − 3.84·39-s + 2.34·41-s − 1.37·43-s − 1.49·45-s − 0.875·47-s + 10/7·49-s + 1.68·51-s − 2.33·53-s − 0.640·61-s − 1.00·63-s + 4.96·65-s − 1.09·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{4} \) |
| 17 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 3 | $C_2 \wr C_2\wr C_2$ | \( 1 - p T + 7 T^{2} - 11 T^{3} + 16 T^{4} - 11 p T^{5} + 7 p^{2} T^{6} - p^{4} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2 \wr C_2\wr C_2$ | \( 1 + p T + 21 T^{2} + 67 T^{3} + 156 T^{4} + 67 p T^{5} + 21 p^{2} T^{6} + p^{4} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 + 16 T^{2} + 142 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 23 | $C_2 \wr C_2\wr C_2$ | \( 1 + 6 T + 72 T^{2} + 286 T^{3} + 2126 T^{4} + 286 p T^{5} + 72 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr C_2\wr C_2$ | \( 1 + 8 T + 84 T^{2} + 504 T^{3} + 3110 T^{4} + 504 p T^{5} + 84 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr C_2\wr C_2$ | \( 1 - 7 T + 21 T^{2} + 229 T^{3} - 2220 T^{4} + 229 p T^{5} + 21 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr C_2\wr C_2$ | \( 1 + 8 T + 144 T^{2} + 808 T^{3} + 7854 T^{4} + 808 p T^{5} + 144 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr C_2\wr C_2$ | \( 1 - 15 T + 177 T^{2} - 1413 T^{3} + 10020 T^{4} - 1413 p T^{5} + 177 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr C_2\wr C_2$ | \( 1 + 9 T + 117 T^{2} + 561 T^{3} + 5196 T^{4} + 561 p T^{5} + 117 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr C_2\wr C_2$ | \( 1 + 6 T + 64 T^{2} + 78 T^{3} + 2238 T^{4} + 78 p T^{5} + 64 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr C_2\wr C_2$ | \( 1 + 17 T + 235 T^{2} + 2023 T^{3} + 16792 T^{4} + 2023 p T^{5} + 235 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 + 124 T^{2} + 8182 T^{4} + 124 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2 \wr C_2\wr C_2$ | \( 1 + 5 T + 245 T^{2} + 907 T^{3} + 22444 T^{4} + 907 p T^{5} + 245 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr C_2\wr C_2$ | \( 1 + 9 T + 215 T^{2} + 1833 T^{3} + 19912 T^{4} + 1833 p T^{5} + 215 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr C_2\wr C_2$ | \( 1 - 12 T + 140 T^{2} - 1148 T^{3} + 7238 T^{4} - 1148 p T^{5} + 140 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr C_2\wr C_2$ | \( 1 + 11 T + 193 T^{2} + 1857 T^{3} + 20804 T^{4} + 1857 p T^{5} + 193 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr C_2\wr C_2$ | \( 1 - 6 T + 192 T^{2} - 654 T^{3} + 18494 T^{4} - 654 p T^{5} + 192 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr C_2\wr C_2$ | \( 1 + 6 T - 554 T^{3} + 686 T^{4} - 554 p T^{5} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr C_2\wr C_2$ | \( 1 - 2 T + 180 T^{2} - 1054 T^{3} + 18198 T^{4} - 1054 p T^{5} + 180 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr C_2\wr C_2$ | \( 1 - 9 T + 265 T^{2} - 1795 T^{3} + 34052 T^{4} - 1795 p T^{5} + 265 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.14771593025336144101653755013, −5.54438612605358544619140777902, −5.37310499371124220508830894406, −5.36209948564928891416979417902, −5.29960369595800868594561887302, −4.91349603746266409927078272020, −4.63101395136686085929129036214, −4.43656470969602166120263086506, −4.30464540395139742565997541484, −4.09096277293632259606783324496, −4.04807239370015839705486618056, −3.75795796875149785005791924896, −3.52715820691068424466392719626, −3.39652196450759112267678827434, −3.25270958787637503703492646918, −2.99673658784287781177664752492, −2.94867858409933930547363617605, −2.76583735985730193786449659680, −2.45137928557803111119186695079, −2.18726745538713429309706840397, −2.09181002117264184599933054698, −1.68442220448626835751466478803, −1.65178277578458985565291360045, −0.986516202633092642417065190050, −0.874295096268019284418240540795, 0, 0, 0, 0,
0.874295096268019284418240540795, 0.986516202633092642417065190050, 1.65178277578458985565291360045, 1.68442220448626835751466478803, 2.09181002117264184599933054698, 2.18726745538713429309706840397, 2.45137928557803111119186695079, 2.76583735985730193786449659680, 2.94867858409933930547363617605, 2.99673658784287781177664752492, 3.25270958787637503703492646918, 3.39652196450759112267678827434, 3.52715820691068424466392719626, 3.75795796875149785005791924896, 4.04807239370015839705486618056, 4.09096277293632259606783324496, 4.30464540395139742565997541484, 4.43656470969602166120263086506, 4.63101395136686085929129036214, 4.91349603746266409927078272020, 5.29960369595800868594561887302, 5.36209948564928891416979417902, 5.37310499371124220508830894406, 5.54438612605358544619140777902, 6.14771593025336144101653755013
