| L(s) = 1 | + 9·3-s + 11·5-s + 7-s + 27·9-s − 19·11-s − 44·13-s + 99·15-s + 104·17-s + 202·19-s + 9·21-s − 280·23-s + 155·25-s − 54·27-s − 146·29-s − 131·31-s − 171·33-s + 11·35-s + 326·37-s − 396·39-s − 1.03e3·41-s − 72·43-s + 297·45-s − 126·47-s + 17·49-s + 936·51-s + 385·53-s − 209·55-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.983·5-s + 0.0539·7-s + 9-s − 0.520·11-s − 0.938·13-s + 1.70·15-s + 1.48·17-s + 2.43·19-s + 0.0935·21-s − 2.53·23-s + 1.23·25-s − 0.384·27-s − 0.934·29-s − 0.758·31-s − 0.902·33-s + 0.0531·35-s + 1.44·37-s − 1.62·39-s − 3.93·41-s − 0.255·43-s + 0.983·45-s − 0.391·47-s + 0.0495·49-s + 2.56·51-s + 0.997·53-s − 0.512·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3436972435\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3436972435\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 - p T + p^{2} T^{2} )^{3} \) |
| 7 | \( 1 - T - 16 T^{2} - 1399 p T^{3} - 16 p^{3} T^{4} - p^{6} T^{5} + p^{9} T^{6} \) |
| good | 5 | \( 1 - 11 T - 34 T^{2} + 3899 T^{3} - 26172 T^{4} - 35987 p T^{5} + 6863704 T^{6} - 35987 p^{4} T^{7} - 26172 p^{6} T^{8} + 3899 p^{9} T^{9} - 34 p^{12} T^{10} - 11 p^{15} T^{11} + p^{18} T^{12} \) |
| 11 | \( 1 + 19 T - 3008 T^{2} - 43697 T^{3} + 5842796 T^{4} + 44696503 T^{5} - 8130449114 T^{6} + 44696503 p^{3} T^{7} + 5842796 p^{6} T^{8} - 43697 p^{9} T^{9} - 3008 p^{12} T^{10} + 19 p^{15} T^{11} + p^{18} T^{12} \) |
| 13 | \( ( 1 + 22 T + 3096 T^{2} + 123644 T^{3} + 3096 p^{3} T^{4} + 22 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 17 | \( 1 - 104 T - 6483 T^{2} + 236648 T^{3} + 108406726 T^{4} - 2440557608 T^{5} - 458074570039 T^{6} - 2440557608 p^{3} T^{7} + 108406726 p^{6} T^{8} + 236648 p^{9} T^{9} - 6483 p^{12} T^{10} - 104 p^{15} T^{11} + p^{18} T^{12} \) |
| 19 | \( 1 - 202 T + 27470 T^{2} - 1603332 T^{3} - 23766086 T^{4} + 20177789350 T^{5} - 2196685326746 T^{6} + 20177789350 p^{3} T^{7} - 23766086 p^{6} T^{8} - 1603332 p^{9} T^{9} + 27470 p^{12} T^{10} - 202 p^{15} T^{11} + p^{18} T^{12} \) |
| 23 | \( 1 + 280 T + 34923 T^{2} + 1611992 T^{3} - 117354650 T^{4} - 25188714152 T^{5} - 3115694739361 T^{6} - 25188714152 p^{3} T^{7} - 117354650 p^{6} T^{8} + 1611992 p^{9} T^{9} + 34923 p^{12} T^{10} + 280 p^{15} T^{11} + p^{18} T^{12} \) |
| 29 | \( ( 1 + 73 T + 44143 T^{2} + 4531786 T^{3} + 44143 p^{3} T^{4} + 73 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 31 | \( 1 + 131 T - 24343 T^{2} - 1860246 T^{3} + 8165167 T^{4} - 56252834717 T^{5} - 356109823730 T^{6} - 56252834717 p^{3} T^{7} + 8165167 p^{6} T^{8} - 1860246 p^{9} T^{9} - 24343 p^{12} T^{10} + 131 p^{15} T^{11} + p^{18} T^{12} \) |
| 37 | \( 1 - 326 T - 40436 T^{2} + 18187808 T^{3} + 3332413048 T^{4} - 833815922582 T^{5} - 13502224437562 T^{6} - 833815922582 p^{3} T^{7} + 3332413048 p^{6} T^{8} + 18187808 p^{9} T^{9} - 40436 p^{12} T^{10} - 326 p^{15} T^{11} + p^{18} T^{12} \) |
| 41 | \( ( 1 + 516 T + 211167 T^{2} + 56124328 T^{3} + 211167 p^{3} T^{4} + 516 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 43 | \( ( 1 + 36 T + 97410 T^{2} - 1479718 T^{3} + 97410 p^{3} T^{4} + 36 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 47 | \( 1 + 126 T - 3099 p T^{2} - 55338442 T^{3} + 4552653426 T^{4} + 3245187461862 T^{5} + 790505587956491 T^{6} + 3245187461862 p^{3} T^{7} + 4552653426 p^{6} T^{8} - 55338442 p^{9} T^{9} - 3099 p^{13} T^{10} + 126 p^{15} T^{11} + p^{18} T^{12} \) |
| 53 | \( 1 - 385 T - 278274 T^{2} + 65424721 T^{3} + 74202339448 T^{4} - 8609250448537 T^{5} - 10584598573216912 T^{6} - 8609250448537 p^{3} T^{7} + 74202339448 p^{6} T^{8} + 65424721 p^{9} T^{9} - 278274 p^{12} T^{10} - 385 p^{15} T^{11} + p^{18} T^{12} \) |
| 59 | \( 1 - 285 T - 484380 T^{2} + 69348171 T^{3} + 170892879300 T^{4} - 11007191102325 T^{5} - 38514898051130018 T^{6} - 11007191102325 p^{3} T^{7} + 170892879300 p^{6} T^{8} + 69348171 p^{9} T^{9} - 484380 p^{12} T^{10} - 285 p^{15} T^{11} + p^{18} T^{12} \) |
| 61 | \( 1 - 34 T - 386391 T^{2} + 62173122 T^{3} + 60945926362 T^{4} - 10370796915418 T^{5} - 9342754356649463 T^{6} - 10370796915418 p^{3} T^{7} + 60945926362 p^{6} T^{8} + 62173122 p^{9} T^{9} - 386391 p^{12} T^{10} - 34 p^{15} T^{11} + p^{18} T^{12} \) |
| 67 | \( 1 + 100 T - 745250 T^{2} + 9713732 T^{3} + 339960743350 T^{4} - 11532432854600 T^{5} - 116964720185345650 T^{6} - 11532432854600 p^{3} T^{7} + 339960743350 p^{6} T^{8} + 9713732 p^{9} T^{9} - 745250 p^{12} T^{10} + 100 p^{15} T^{11} + p^{18} T^{12} \) |
| 71 | \( ( 1 + 34 T + 294609 T^{2} - 182711756 T^{3} + 294609 p^{3} T^{4} + 34 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 73 | \( 1 - 108 T - 177048 T^{2} - 671641700 T^{3} - 2631873744 T^{4} + 60963298021476 T^{5} + 208159411130844486 T^{6} + 60963298021476 p^{3} T^{7} - 2631873744 p^{6} T^{8} - 671641700 p^{9} T^{9} - 177048 p^{12} T^{10} - 108 p^{15} T^{11} + p^{18} T^{12} \) |
| 79 | \( 1 + 2463 T + 2638029 T^{2} + 2601019670 T^{3} + 2867577805215 T^{4} + 2288144397114963 T^{5} + 1505212561410160614 T^{6} + 2288144397114963 p^{3} T^{7} + 2867577805215 p^{6} T^{8} + 2601019670 p^{9} T^{9} + 2638029 p^{12} T^{10} + 2463 p^{15} T^{11} + p^{18} T^{12} \) |
| 83 | \( ( 1 - 115 T + 1228989 T^{2} - 243220678 T^{3} + 1228989 p^{3} T^{4} - 115 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 89 | \( 1 - 110 T - 375783 T^{2} - 427037698 T^{3} - 101851887698 T^{4} + 80035470446002 T^{5} + 625569659939630333 T^{6} + 80035470446002 p^{3} T^{7} - 101851887698 p^{6} T^{8} - 427037698 p^{9} T^{9} - 375783 p^{12} T^{10} - 110 p^{15} T^{11} + p^{18} T^{12} \) |
| 97 | \( ( 1 + 2941 T + 5549515 T^{2} + 6235037870 T^{3} + 5549515 p^{3} T^{4} + 2941 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.65442511572058073763705673474, −5.64638934689864623197982019901, −5.53766098426031152498719381985, −5.38225007507975364742087632458, −5.10796200301954845500892144145, −4.84619434292691606971374937160, −4.77165017387606705044821657011, −4.46678617279335676198333375108, −4.15990444283743466823064403846, −3.98351915981060369519610405419, −3.82171439316299136748098236939, −3.42937756663863946666132343063, −3.34241501858837087071526852637, −3.19599406686120794883745832941, −3.13747048269293780862012571525, −2.58732358850318283383138095580, −2.58477432346666934109672244112, −2.28929889568573236395002451326, −2.23755259585099508961440100353, −1.55645030340641848042154032163, −1.52808843931193865042588283387, −1.51301850842592350430045237828, −1.06659293848363275262715768338, −0.50125577844024917112917132801, −0.04963998540025867933547828281,
0.04963998540025867933547828281, 0.50125577844024917112917132801, 1.06659293848363275262715768338, 1.51301850842592350430045237828, 1.52808843931193865042588283387, 1.55645030340641848042154032163, 2.23755259585099508961440100353, 2.28929889568573236395002451326, 2.58477432346666934109672244112, 2.58732358850318283383138095580, 3.13747048269293780862012571525, 3.19599406686120794883745832941, 3.34241501858837087071526852637, 3.42937756663863946666132343063, 3.82171439316299136748098236939, 3.98351915981060369519610405419, 4.15990444283743466823064403846, 4.46678617279335676198333375108, 4.77165017387606705044821657011, 4.84619434292691606971374937160, 5.10796200301954845500892144145, 5.38225007507975364742087632458, 5.53766098426031152498719381985, 5.64638934689864623197982019901, 5.65442511572058073763705673474
Plot not available for L-functions of degree greater than 10.
