| 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
| \(L(1)\) |
\(\approx\) |
\(16.42014991\) |
| \(L(\frac12)\) |
\(\approx\) |
\(16.42014991\) |
| \(L(\frac{3}{2})\) |
|
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
−7.32458156123370363064492509504, −6.94229084198360791743655812930, −6.76564135494387278583369946505, −6.62036679241614847665031389416, −6.08221945284508279113644741136, −6.03640580424431193897516602648, −5.69324566495061314692373011117, −5.09325629723292996427698473080, −5.08634210236837009596409191762, −4.92471754280951581694322969201, −4.50900563267471259291440075121, −4.38110104469160382783253654372, −4.32859579416541790719222521348, −3.67741972315176807732138226514, −3.56902569332840156590365931314, −3.52185490773513273936262566005, −2.87959091015413145876618168068, −2.81429257764261978914754210973, −2.79660872319380591250899830837, −2.00662359035943474790942900837, −1.85472130370154996300596592429, −1.73450093400169879368293793299, −1.00176610861547276937527916582, −0.869081742809182179977270810884, −0.67878950524728812232264579612,
0.67878950524728812232264579612, 0.869081742809182179977270810884, 1.00176610861547276937527916582, 1.73450093400169879368293793299, 1.85472130370154996300596592429, 2.00662359035943474790942900837, 2.79660872319380591250899830837, 2.81429257764261978914754210973, 2.87959091015413145876618168068, 3.52185490773513273936262566005, 3.56902569332840156590365931314, 3.67741972315176807732138226514, 4.32859579416541790719222521348, 4.38110104469160382783253654372, 4.50900563267471259291440075121, 4.92471754280951581694322969201, 5.08634210236837009596409191762, 5.09325629723292996427698473080, 5.69324566495061314692373011117, 6.03640580424431193897516602648, 6.08221945284508279113644741136, 6.62036679241614847665031389416, 6.76564135494387278583369946505, 6.94229084198360791743655812930, 7.32458156123370363064492509504
