| L(s) = 1 | − 3·3-s − 3·5-s − 5·7-s + 6·9-s + 11-s + 3·13-s + 9·15-s + 17-s − 4·19-s + 15·21-s − 3·23-s + 6·25-s − 10·27-s + 2·29-s − 12·31-s − 3·33-s + 15·35-s + 7·37-s − 9·39-s − 5·41-s + 2·43-s − 18·45-s − 4·47-s + 2·49-s − 3·51-s + 3·53-s − 3·55-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 1.34·5-s − 1.88·7-s + 2·9-s + 0.301·11-s + 0.832·13-s + 2.32·15-s + 0.242·17-s − 0.917·19-s + 3.27·21-s − 0.625·23-s + 6/5·25-s − 1.92·27-s + 0.371·29-s − 2.15·31-s − 0.522·33-s + 2.53·35-s + 1.15·37-s − 1.44·39-s − 0.780·41-s + 0.304·43-s − 2.68·45-s − 0.583·47-s + 2/7·49-s − 0.420·51-s + 0.412·53-s − 0.404·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 3^{3} \cdot 5^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 3^{3} \cdot 5^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5503566316\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5503566316\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{3} \) |
| 5 | $C_1$ | \( ( 1 + T )^{3} \) |
| 13 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 7 | $S_4\times C_2$ | \( 1 + 5 T + 23 T^{2} + 66 T^{3} + 23 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - T + 21 T^{2} - 6 T^{3} + 21 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - T + T^{2} + 114 T^{3} + p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 4 T + 41 T^{2} + 96 T^{3} + 41 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 3 T + 15 T^{2} + 134 T^{3} + 15 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 2 T + 27 T^{2} + 108 T^{3} + 27 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{3} \) |
| 37 | $S_4\times C_2$ | \( 1 - 7 T + 115 T^{2} - 490 T^{3} + 115 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 5 T + 119 T^{2} + 406 T^{3} + 119 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 2 T + 81 T^{2} - 44 T^{3} + 81 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 4 T + 45 T^{2} + 120 T^{3} + 45 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 3 T + 51 T^{2} - 10 p T^{3} + 51 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 113 T^{2} - 64 T^{3} + 113 p T^{4} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 3 T + 143 T^{2} + 218 T^{3} + 143 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 71 | $S_4\times C_2$ | \( 1 + 11 T + 221 T^{2} + 1546 T^{3} + 221 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 4 T + 67 T^{2} + 320 T^{3} + 67 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 25 T + 413 T^{2} + 4206 T^{3} + 413 p T^{4} + 25 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 20 T + 281 T^{2} - 3064 T^{3} + 281 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 7 T + 251 T^{2} + 1130 T^{3} + 251 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 23 T + 461 T^{2} - 4866 T^{3} + 461 p T^{4} - 23 p^{2} T^{5} + p^{3} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.03736772442931028999836293926, −6.76394350716210226102843268728, −6.64366653482579490182595855018, −6.36576272413997351637909058476, −6.09679537805940822595706969829, −5.99294247676327140350225776342, −5.94424414758466026745882862043, −5.42048069451708364182279713237, −5.13556563300578095314200385374, −5.11317096441213363646525872820, −4.52982785267199825028336350513, −4.50141902085772549251461740741, −4.12110976583487756346814198119, −3.80100929852551537698359533036, −3.73739995862293131032922008528, −3.62872994042366668895248465251, −2.96327305345754045850621035616, −2.94608010752171278207937495414, −2.73324285196159800349578383176, −1.88980673205517871866056331925, −1.73997531847037311384829487592, −1.52064480685058520221892285901, −0.817351016587291597509076379066, −0.42265836307824378590430918545, −0.34177022352854767219290519722,
0.34177022352854767219290519722, 0.42265836307824378590430918545, 0.817351016587291597509076379066, 1.52064480685058520221892285901, 1.73997531847037311384829487592, 1.88980673205517871866056331925, 2.73324285196159800349578383176, 2.94608010752171278207937495414, 2.96327305345754045850621035616, 3.62872994042366668895248465251, 3.73739995862293131032922008528, 3.80100929852551537698359533036, 4.12110976583487756346814198119, 4.50141902085772549251461740741, 4.52982785267199825028336350513, 5.11317096441213363646525872820, 5.13556563300578095314200385374, 5.42048069451708364182279713237, 5.94424414758466026745882862043, 5.99294247676327140350225776342, 6.09679537805940822595706969829, 6.36576272413997351637909058476, 6.64366653482579490182595855018, 6.76394350716210226102843268728, 7.03736772442931028999836293926
