| L(s) = 1 | + 9·3-s − 4·5-s − 30·7-s + 54·9-s − 16·11-s − 39·13-s − 36·15-s − 146·17-s + 94·19-s − 270·21-s + 48·23-s − 107·25-s + 270·27-s + 2·29-s − 302·31-s − 144·33-s + 120·35-s − 374·37-s − 351·39-s + 480·41-s − 260·43-s − 216·45-s + 24·47-s + 159·49-s − 1.31e3·51-s + 678·53-s + 64·55-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 0.357·5-s − 1.61·7-s + 2·9-s − 0.438·11-s − 0.832·13-s − 0.619·15-s − 2.08·17-s + 1.13·19-s − 2.80·21-s + 0.435·23-s − 0.855·25-s + 1.92·27-s + 0.0128·29-s − 1.74·31-s − 0.759·33-s + 0.579·35-s − 1.66·37-s − 1.44·39-s + 1.82·41-s − 0.922·43-s − 0.715·45-s + 0.0744·47-s + 0.463·49-s − 3.60·51-s + 1.75·53-s + 0.156·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7615679776\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7615679776\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{3} \) |
| 13 | $C_1$ | \( ( 1 + p T )^{3} \) |
| good | 5 | $S_4\times C_2$ | \( 1 + 4 T + 123 T^{2} + 1864 T^{3} + 123 p^{3} T^{4} + 4 p^{6} T^{5} + p^{9} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 30 T + 741 T^{2} + 18596 T^{3} + 741 p^{3} T^{4} + 30 p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 16 T + 1737 T^{2} + 72928 T^{3} + 1737 p^{3} T^{4} + 16 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 146 T + 20799 T^{2} + 1505852 T^{3} + 20799 p^{3} T^{4} + 146 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 94 T + 6145 T^{2} - 509876 T^{3} + 6145 p^{3} T^{4} - 94 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 48 T + 15573 T^{2} - 1702560 T^{3} + 15573 p^{3} T^{4} - 48 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 2 T + 63051 T^{2} + 101620 T^{3} + 63051 p^{3} T^{4} - 2 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 302 T + 71837 T^{2} + 10796516 T^{3} + 71837 p^{3} T^{4} + 302 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 374 T + 114995 T^{2} + 30130340 T^{3} + 114995 p^{3} T^{4} + 374 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 480 T + 208479 T^{2} - 53244336 T^{3} + 208479 p^{3} T^{4} - 480 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 260 T + 200425 T^{2} + 37680472 T^{3} + 200425 p^{3} T^{4} + 260 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 24 T + 142989 T^{2} - 23086032 T^{3} + 142989 p^{3} T^{4} - 24 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 678 T + 404403 T^{2} - 200405604 T^{3} + 404403 p^{3} T^{4} - 678 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 1788 T + 1572249 T^{2} + 871859112 T^{3} + 1572249 p^{3} T^{4} + 1788 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 230 T + 636491 T^{2} + 98131748 T^{3} + 636491 p^{3} T^{4} + 230 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 74 T + 493073 T^{2} - 40252028 T^{3} + 493073 p^{3} T^{4} - 74 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 948 T + 1061157 T^{2} - 608134872 T^{3} + 1061157 p^{3} T^{4} - 948 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 222 T + 223815 T^{2} + 195504100 T^{3} + 223815 p^{3} T^{4} + 222 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 24 T + 1400781 T^{2} - 31423696 T^{3} + 1400781 p^{3} T^{4} - 24 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 796 T + 1904433 T^{2} + 924248872 T^{3} + 1904433 p^{3} T^{4} + 796 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 1436 T + 2536191 T^{2} - 2054800856 T^{3} + 2536191 p^{3} T^{4} - 1436 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 3242 T + 6203519 T^{2} - 7136252780 T^{3} + 6203519 p^{3} T^{4} - 3242 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−7.59420983018327530720123366028, −7.29711911009943972386174297877, −7.21529788314404275732040974071, −6.92632869987344471192449777976, −6.54159351642870121880040582389, −6.34599788074816316542741184182, −6.20716719136986701666707093458, −5.69665496249946830159524220028, −5.40921095693756550761939167579, −5.15590519515866338054944612797, −4.71538932789604488522874336505, −4.53482624061084427885436829350, −4.33766643276521165193592367219, −3.65803476191574603883679481209, −3.63195102712601588450473728326, −3.60379449808299288755236797723, −2.97095673881482959713375786822, −2.80525834452550009861043203213, −2.72171525885113310138879275239, −1.99977560958839252735703784411, −1.94965332031322184640039832413, −1.75476383729905286427139137374, −1.00196879915714256045522685230, −0.52130919145698112017973776387, −0.12527939187395883882478731892,
0.12527939187395883882478731892, 0.52130919145698112017973776387, 1.00196879915714256045522685230, 1.75476383729905286427139137374, 1.94965332031322184640039832413, 1.99977560958839252735703784411, 2.72171525885113310138879275239, 2.80525834452550009861043203213, 2.97095673881482959713375786822, 3.60379449808299288755236797723, 3.63195102712601588450473728326, 3.65803476191574603883679481209, 4.33766643276521165193592367219, 4.53482624061084427885436829350, 4.71538932789604488522874336505, 5.15590519515866338054944612797, 5.40921095693756550761939167579, 5.69665496249946830159524220028, 6.20716719136986701666707093458, 6.34599788074816316542741184182, 6.54159351642870121880040582389, 6.92632869987344471192449777976, 7.21529788314404275732040974071, 7.29711911009943972386174297877, 7.59420983018327530720123366028
