| L(s) = 1 | + 2·2-s − 3·3-s + 4-s − 6·6-s + 3·7-s − 8-s + 6·9-s + 8·11-s − 3·12-s + 6·14-s − 5·16-s + 4·17-s + 12·18-s + 7·19-s − 9·21-s + 16·22-s + 9·23-s + 3·24-s − 2·25-s − 10·27-s + 3·28-s − 7·29-s + 7·31-s − 8·32-s − 24·33-s + 8·34-s + 6·36-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.73·3-s + 1/2·4-s − 2.44·6-s + 1.13·7-s − 0.353·8-s + 2·9-s + 2.41·11-s − 0.866·12-s + 1.60·14-s − 5/4·16-s + 0.970·17-s + 2.82·18-s + 1.60·19-s − 1.96·21-s + 3.41·22-s + 1.87·23-s + 0.612·24-s − 2/5·25-s − 1.92·27-s + 0.566·28-s − 1.29·29-s + 1.25·31-s − 1.41·32-s − 4.17·33-s + 1.37·34-s + 36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 7^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 7^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.401632500\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.401632500\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 + T )^{3} \) |
| 7 | $C_1$ | \( ( 1 - T )^{3} \) |
| 13 | | \( 1 \) |
| good | 2 | $A_4\times C_2$ | \( 1 - p T + 3 T^{2} - 3 T^{3} + 3 p T^{4} - p^{3} T^{5} + p^{3} T^{6} \) |
| 5 | $A_4\times C_2$ | \( 1 + 2 T^{2} - 13 T^{3} + 2 p T^{4} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 - 8 T + 50 T^{2} - 181 T^{3} + 50 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 - 4 T + 52 T^{2} - 135 T^{3} + 52 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 - 7 T + 56 T^{2} - 219 T^{3} + 56 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 - 9 T + 83 T^{2} - 415 T^{3} + 83 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 + 7 T + 73 T^{2} + 411 T^{3} + 73 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 - 7 T + 53 T^{2} - 153 T^{3} + 53 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 + 98 T^{2} + 13 T^{3} + 98 p T^{4} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 + 2 T + 55 T^{2} - 36 T^{3} + 55 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 + 19 T + 245 T^{2} + 1863 T^{3} + 245 p T^{4} + 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 - 17 T + 155 T^{2} - 1051 T^{3} + 155 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 - 13 T + 198 T^{2} - 1365 T^{3} + 198 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 - 3 T + 167 T^{2} - 329 T^{3} + 167 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 - 13 T + 222 T^{2} - 1573 T^{3} + 222 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 + 5 T + 127 T^{2} + 275 T^{3} + 127 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 + 8 T + 100 T^{2} + 621 T^{3} + 100 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - 2 T + 138 T^{2} - 5 p T^{3} + 138 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + T - 53 T^{2} - 179 T^{3} - 53 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 + 2 T + 64 T^{2} + 561 T^{3} + 64 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 - 19 T + 266 T^{2} - 2363 T^{3} + 266 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 + 27 T + 521 T^{2} + 5837 T^{3} + 521 p T^{4} + 27 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.35042526427130675602877750286, −7.04802195279828171736019572459, −6.97027882700484109820538438746, −6.78006506092938548318165966703, −6.70301662142468319904745153970, −6.23434365851683346155090671354, −5.79151973835157365316797329305, −5.65816700240242021542346142750, −5.56618559605753250548155602929, −5.34845291831151937980773952543, −5.00260291304655767531995879396, −4.77793063202323496371957937415, −4.61869339580710429202409874929, −4.06432155851238166227655119491, −4.05410010907938555991020874017, −4.05351641740898799198810226947, −3.36468672636334245559103955998, −3.19352412319857796456294729873, −3.02245943782334337486286715960, −2.15698959497031289396051127317, −2.06448737829199994063437093168, −1.46949624989147591450027483292, −1.27680981914522532426862860838, −0.793764586305138567554469106742, −0.65879586167169024558920729867,
0.65879586167169024558920729867, 0.793764586305138567554469106742, 1.27680981914522532426862860838, 1.46949624989147591450027483292, 2.06448737829199994063437093168, 2.15698959497031289396051127317, 3.02245943782334337486286715960, 3.19352412319857796456294729873, 3.36468672636334245559103955998, 4.05351641740898799198810226947, 4.05410010907938555991020874017, 4.06432155851238166227655119491, 4.61869339580710429202409874929, 4.77793063202323496371957937415, 5.00260291304655767531995879396, 5.34845291831151937980773952543, 5.56618559605753250548155602929, 5.65816700240242021542346142750, 5.79151973835157365316797329305, 6.23434365851683346155090671354, 6.70301662142468319904745153970, 6.78006506092938548318165966703, 6.97027882700484109820538438746, 7.04802195279828171736019572459, 7.35042526427130675602877750286
