| L(s) = 1 | − 3·3-s − 3·5-s + 2·7-s + 6·9-s − 4·11-s − 2·13-s + 9·15-s + 6·17-s − 2·19-s − 6·21-s − 3·23-s + 6·25-s − 10·27-s − 2·31-s + 12·33-s − 6·35-s + 8·37-s + 6·39-s + 2·41-s − 10·43-s − 18·45-s − 6·47-s − 6·49-s − 18·51-s + 6·53-s + 12·55-s + 6·57-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 1.34·5-s + 0.755·7-s + 2·9-s − 1.20·11-s − 0.554·13-s + 2.32·15-s + 1.45·17-s − 0.458·19-s − 1.30·21-s − 0.625·23-s + 6/5·25-s − 1.92·27-s − 0.359·31-s + 2.08·33-s − 1.01·35-s + 1.31·37-s + 0.960·39-s + 0.312·41-s − 1.52·43-s − 2.68·45-s − 0.875·47-s − 6/7·49-s − 2.52·51-s + 0.824·53-s + 1.61·55-s + 0.794·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{3} \) |
| 5 | $C_1$ | \( ( 1 + T )^{3} \) |
| 23 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 7 | $S_4\times C_2$ | \( 1 - 2 T + 10 T^{2} - 12 T^{3} + 10 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 4 T + 23 T^{2} + 52 T^{3} + 23 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 2 T + 25 T^{2} + 60 T^{3} + 25 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 6 T + 52 T^{2} - 178 T^{3} + 52 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 2 T + 43 T^{2} + 84 T^{3} + 43 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 76 T^{2} + 12 T^{3} + 76 p T^{4} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 2 T + 82 T^{2} + 108 T^{3} + 82 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 8 T + 60 T^{2} - 378 T^{3} + 60 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 2 T + 24 T^{2} - 2 p T^{3} + 24 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 10 T + 101 T^{2} + 836 T^{3} + 101 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 6 T + 7 T^{2} - 372 T^{3} + 7 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 6 T + 72 T^{2} - 122 T^{3} + 72 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 8 T + 122 T^{2} + 1016 T^{3} + 122 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 4 T + 173 T^{2} - 452 T^{3} + 173 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 8 T + 150 T^{2} + 858 T^{3} + 150 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 18 T + 310 T^{2} + 2694 T^{3} + 310 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 8 T + 153 T^{2} - 684 T^{3} + 153 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{3} \) |
| 83 | $S_4\times C_2$ | \( 1 + 24 T + 430 T^{2} + 4420 T^{3} + 430 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 4 T + 147 T^{2} - 136 T^{3} + 147 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 12 T + 163 T^{2} - 2456 T^{3} + 163 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.66845488519256044728747410436, −7.24367005002506227935140667681, −7.09436835935119714163296833583, −7.03347556699167426678120266592, −6.50881531120310829323965425972, −6.32069073048154739223890032847, −6.20166944138154289931531452790, −5.68501694194398970661644220707, −5.68131607578550732693994556249, −5.42898443117621763757481321457, −5.00604404174352767014504480294, −4.87711453049120629234004679352, −4.84120780173396783169877542167, −4.31605340502371419552091819719, −4.29056185448090099241933383947, −4.01870177500749694701357688897, −3.40727314929811459570682418444, −3.31266233979619403431868187756, −3.27927313281922929656965694222, −2.52075107973109547310492990921, −2.35055224617613829107549962099, −2.11005449893527353981997735215, −1.37710512594548165282861031508, −1.20247364932587993758192760713, −1.05770302617240638448708093010, 0, 0, 0,
1.05770302617240638448708093010, 1.20247364932587993758192760713, 1.37710512594548165282861031508, 2.11005449893527353981997735215, 2.35055224617613829107549962099, 2.52075107973109547310492990921, 3.27927313281922929656965694222, 3.31266233979619403431868187756, 3.40727314929811459570682418444, 4.01870177500749694701357688897, 4.29056185448090099241933383947, 4.31605340502371419552091819719, 4.84120780173396783169877542167, 4.87711453049120629234004679352, 5.00604404174352767014504480294, 5.42898443117621763757481321457, 5.68131607578550732693994556249, 5.68501694194398970661644220707, 6.20166944138154289931531452790, 6.32069073048154739223890032847, 6.50881531120310829323965425972, 7.03347556699167426678120266592, 7.09436835935119714163296833583, 7.24367005002506227935140667681, 7.66845488519256044728747410436
