| L(s) = 1 | + 4·2-s + 54·3-s − 212·4-s + 250·5-s + 216·6-s − 952·7-s − 1.24e3·8-s + 2.18e3·9-s + 1.00e3·10-s + 2.66e3·11-s − 1.14e4·12-s − 1.04e3·13-s − 3.80e3·14-s + 1.35e4·15-s + 2.90e4·16-s − 1.09e4·17-s + 8.74e3·18-s − 8.14e3·19-s − 5.30e4·20-s − 5.14e4·21-s + 1.06e4·22-s − 1.63e4·23-s − 6.73e4·24-s + 4.68e4·25-s − 4.17e3·26-s + 7.87e4·27-s + 2.01e5·28-s + ⋯ |
| L(s) = 1 | + 0.353·2-s + 1.15·3-s − 1.65·4-s + 0.894·5-s + 0.408·6-s − 1.04·7-s − 0.861·8-s + 9-s + 0.316·10-s + 0.603·11-s − 1.91·12-s − 0.131·13-s − 0.370·14-s + 1.03·15-s + 1.77·16-s − 0.541·17-s + 0.353·18-s − 0.272·19-s − 1.48·20-s − 1.21·21-s + 0.213·22-s − 0.279·23-s − 0.995·24-s + 3/5·25-s − 0.0465·26-s + 0.769·27-s + 1.73·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27225 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - p^{2} T + 57 p^{2} T^{2} - p^{9} T^{3} + p^{14} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 136 p T + 1369654 T^{2} + 136 p^{8} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 1044 T + 51398310 T^{2} + 1044 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 10964 T + 780399870 T^{2} + 10964 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 8144 T + 1803193654 T^{2} + 8144 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 16304 T + 6573525998 T^{2} + 16304 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 168748 T + 40054753902 T^{2} + 168748 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 168240 T + 51688425950 T^{2} + 168240 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 138612 T + 178918920510 T^{2} + 138612 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 678164 T + 431361092598 T^{2} + 678164 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 234816 T + 358323692286 T^{2} - 234816 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 820976 T + 1167879519422 T^{2} + 820976 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 1703956 T + 2047535212350 T^{2} + 1703956 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 1520648 T + 5170129134614 T^{2} + 1520648 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 1208484 T + 6633037684878 T^{2} + 1208484 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 1391208 T + 5369867109894 T^{2} + 1391208 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 1432304 T + 18650783996814 T^{2} + 1432304 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 541332 T + 21627169758222 T^{2} - 541332 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 5212920 T + 20024616631406 T^{2} - 5212920 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4155128 T + 54264139490558 T^{2} + 4155128 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 6089788 T + 78760855020246 T^{2} + 6089788 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 25442556 T + 323302662844710 T^{2} + 25442556 p^{7} T^{3} + p^{14} T^{4} \) |
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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
−11.12012931734855909455439373507, −10.61591945495141985086271970783, −9.753101330633221875434041233920, −9.699964515186045556275366133538, −9.144020509029690561718622329145, −9.085349000964970277232203784799, −8.214994244643670745537805873886, −7.961976401624391119397576021955, −6.81153346814317213968074827160, −6.65788437694438405576224506628, −5.69909659189041104973759593951, −5.32690806590723024426530995668, −4.41387207594492839084198511220, −4.11913783583993229043726334300, −3.17903295598457150861524390063, −3.15863091799184086677548632512, −1.87579693091905128199886779802, −1.43930413588224172938049720639, 0, 0,
1.43930413588224172938049720639, 1.87579693091905128199886779802, 3.15863091799184086677548632512, 3.17903295598457150861524390063, 4.11913783583993229043726334300, 4.41387207594492839084198511220, 5.32690806590723024426530995668, 5.69909659189041104973759593951, 6.65788437694438405576224506628, 6.81153346814317213968074827160, 7.961976401624391119397576021955, 8.214994244643670745537805873886, 9.085349000964970277232203784799, 9.144020509029690561718622329145, 9.699964515186045556275366133538, 9.753101330633221875434041233920, 10.61591945495141985086271970783, 11.12012931734855909455439373507
