| L(s) = 1 | − 2·2-s + 10·4-s + 28·7-s + 8·8-s − 56·14-s + 12·16-s − 52·17-s − 328·23-s + 322·25-s + 280·28-s − 636·31-s + 200·32-s + 104·34-s − 236·41-s + 656·46-s + 408·47-s − 310·49-s − 644·50-s + 224·56-s + 1.27e3·62-s + 216·64-s − 520·68-s + 1.70e3·71-s + 956·73-s − 44·79-s + 472·82-s + 220·89-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 5/4·4-s + 1.51·7-s + 0.353·8-s − 1.06·14-s + 3/16·16-s − 0.741·17-s − 2.97·23-s + 2.57·25-s + 1.88·28-s − 3.68·31-s + 1.10·32-s + 0.524·34-s − 0.898·41-s + 2.10·46-s + 1.26·47-s − 0.903·49-s − 1.82·50-s + 0.534·56-s + 2.60·62-s + 0.421·64-s − 0.927·68-s + 2.84·71-s + 1.53·73-s − 0.0626·79-s + 0.635·82-s + 0.262·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.717206887\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.717206887\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T - 3 p T^{2} - 5 p^{3} T^{3} - 3 p^{4} T^{4} + p^{7} T^{5} + p^{9} T^{6} \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 322 T^{2} + 49351 T^{4} - 6170684 T^{6} + 49351 p^{6} T^{8} - 322 p^{12} T^{10} + p^{18} T^{12} \) |
| 7 | \( ( 1 - 2 p T + 449 T^{2} - 3788 T^{3} + 449 p^{3} T^{4} - 2 p^{7} T^{5} + p^{9} T^{6} )^{2} \) |
| 11 | \( 1 - 214 p T^{2} + 37343 p^{2} T^{4} - 5987722076 T^{6} + 37343 p^{8} T^{8} - 214 p^{13} T^{10} + p^{18} T^{12} \) |
| 13 | \( 1 - 8270 T^{2} + 36264983 T^{4} - 97600232804 T^{6} + 36264983 p^{6} T^{8} - 8270 p^{12} T^{10} + p^{18} T^{12} \) |
| 17 | \( ( 1 + 26 T + 3615 T^{2} - 222100 T^{3} + 3615 p^{3} T^{4} + 26 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 19 | \( 1 - 18194 T^{2} + 9648173 p T^{4} - 1372700323292 T^{6} + 9648173 p^{7} T^{8} - 18194 p^{12} T^{10} + p^{18} T^{12} \) |
| 23 | \( ( 1 + 164 T + 42885 T^{2} + 4036280 T^{3} + 42885 p^{3} T^{4} + 164 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 29 | \( 1 - 123986 T^{2} + 6779237687 T^{4} - 212188653261788 T^{6} + 6779237687 p^{6} T^{8} - 123986 p^{12} T^{10} + p^{18} T^{12} \) |
| 31 | \( ( 1 + 318 T + 93849 T^{2} + 15197452 T^{3} + 93849 p^{3} T^{4} + 318 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 37 | \( 1 - 124142 T^{2} + 4233590471 T^{4} - 51775900405988 T^{6} + 4233590471 p^{6} T^{8} - 124142 p^{12} T^{10} + p^{18} T^{12} \) |
| 41 | \( ( 1 + 118 T + 89463 T^{2} - 3720620 T^{3} + 89463 p^{3} T^{4} + 118 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 43 | \( 1 - 247490 T^{2} + 32846327015 T^{4} - 3025278566260412 T^{6} + 32846327015 p^{6} T^{8} - 247490 p^{12} T^{10} + p^{18} T^{12} \) |
| 47 | \( ( 1 - 204 T + 283677 T^{2} - 40395048 T^{3} + 283677 p^{3} T^{4} - 204 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 53 | \( 1 - 292802 T^{2} + 260255467 p T^{4} + 2699981198933572 T^{6} + 260255467 p^{7} T^{8} - 292802 p^{12} T^{10} + p^{18} T^{12} \) |
| 59 | \( 1 - 1093858 T^{2} + 524838290887 T^{4} - 140555838506313212 T^{6} + 524838290887 p^{6} T^{8} - 1093858 p^{12} T^{10} + p^{18} T^{12} \) |
| 61 | \( 1 - 459870 T^{2} + 162687674679 T^{4} - 38865284671151684 T^{6} + 162687674679 p^{6} T^{8} - 459870 p^{12} T^{10} + p^{18} T^{12} \) |
| 67 | \( 1 - 750066 T^{2} + 226548162807 T^{4} - 53949461413257884 T^{6} + 226548162807 p^{6} T^{8} - 750066 p^{12} T^{10} + p^{18} T^{12} \) |
| 71 | \( ( 1 - 12 p T + 1006773 T^{2} - 524795352 T^{3} + 1006773 p^{3} T^{4} - 12 p^{7} T^{5} + p^{9} T^{6} )^{2} \) |
| 73 | \( ( 1 - 478 T + 911095 T^{2} - 251066948 T^{3} + 911095 p^{3} T^{4} - 478 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 79 | \( ( 1 + 22 T + 1407593 T^{2} + 13791100 T^{3} + 1407593 p^{3} T^{4} + 22 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 83 | \( 1 - 2910274 T^{2} + 3775777045015 T^{4} - 2787348361783974908 T^{6} + 3775777045015 p^{6} T^{8} - 2910274 p^{12} T^{10} + p^{18} T^{12} \) |
| 89 | \( ( 1 - 110 T + 2073543 T^{2} - 156516836 T^{3} + 2073543 p^{3} T^{4} - 110 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 97 | \( ( 1 + 1222 T + 2989679 T^{2} + 2155770388 T^{3} + 2989679 p^{3} T^{4} + 1222 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.045645377179463290449833295468, −7.49263734511130003915739014109, −7.41059753212885804327278442272, −7.30258901107697362402404850872, −6.75563828218632858940465637041, −6.73041038624901430387064650907, −6.64588220966801780596876760543, −6.50934330736007173862984076548, −5.74378298449712357146918945872, −5.66844225143501545742768067563, −5.54197876376161718312542982770, −5.23010669682858826769854468766, −4.98395312422848656000627081565, −4.59602182434269913250637073494, −4.42364770896817431671164723413, −4.03113146933281725501759567018, −3.81609962226723954690759502243, −3.50095921147831735041524152822, −3.06755924922705832695293758966, −2.50770158075500088906091389333, −2.17378473121149874479210875903, −1.83479725650786765735280718315, −1.69436309691666560893274177880, −1.20383057377476848954066687950, −0.40238822450246262615746168383,
0.40238822450246262615746168383, 1.20383057377476848954066687950, 1.69436309691666560893274177880, 1.83479725650786765735280718315, 2.17378473121149874479210875903, 2.50770158075500088906091389333, 3.06755924922705832695293758966, 3.50095921147831735041524152822, 3.81609962226723954690759502243, 4.03113146933281725501759567018, 4.42364770896817431671164723413, 4.59602182434269913250637073494, 4.98395312422848656000627081565, 5.23010669682858826769854468766, 5.54197876376161718312542982770, 5.66844225143501545742768067563, 5.74378298449712357146918945872, 6.50934330736007173862984076548, 6.64588220966801780596876760543, 6.73041038624901430387064650907, 6.75563828218632858940465637041, 7.30258901107697362402404850872, 7.41059753212885804327278442272, 7.49263734511130003915739014109, 8.045645377179463290449833295468
Plot not available for L-functions of degree greater than 10.
