| L(s) = 1 | − 2·2-s + 9·3-s + 5·4-s − 15·5-s − 18·6-s + 10·7-s + 4·8-s + 54·9-s + 30·10-s + 33·11-s + 45·12-s + 114·13-s − 20·14-s − 135·15-s − 27·16-s − 104·17-s − 108·18-s − 58·19-s − 75·20-s + 90·21-s − 66·22-s + 120·23-s + 36·24-s + 150·25-s − 228·26-s + 270·27-s + 50·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.73·3-s + 5/8·4-s − 1.34·5-s − 1.22·6-s + 0.539·7-s + 0.176·8-s + 2·9-s + 0.948·10-s + 0.904·11-s + 1.08·12-s + 2.43·13-s − 0.381·14-s − 2.32·15-s − 0.421·16-s − 1.48·17-s − 1.41·18-s − 0.700·19-s − 0.838·20-s + 0.935·21-s − 0.639·22-s + 1.08·23-s + 0.306·24-s + 6/5·25-s − 1.71·26-s + 1.92·27-s + 0.337·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4492125 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4492125 ^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.507126846\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.507126846\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - p T )^{3} \) |
| 5 | $C_1$ | \( ( 1 + p T )^{3} \) |
| 11 | $C_1$ | \( ( 1 - p T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 + p T - T^{2} - p^{4} T^{3} - p^{3} T^{4} + p^{7} T^{5} + p^{9} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 10 T + 425 T^{2} - 3412 T^{3} + 425 p^{3} T^{4} - 10 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 114 T + 10303 T^{2} - 538132 T^{3} + 10303 p^{3} T^{4} - 114 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 104 T + 17531 T^{2} + 1030352 T^{3} + 17531 p^{3} T^{4} + 104 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 58 T + 9081 T^{2} + 861164 T^{3} + 9081 p^{3} T^{4} + 58 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 120 T + 25765 T^{2} - 2771856 T^{3} + 25765 p^{3} T^{4} - 120 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 220 T + 58859 T^{2} + 10101400 T^{3} + 58859 p^{3} T^{4} + 220 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 8 p T + 50781 T^{2} - 5187088 T^{3} + 50781 p^{3} T^{4} - 8 p^{7} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 838 T + 375507 T^{2} - 103501764 T^{3} + 375507 p^{3} T^{4} - 838 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 156 T + 100167 T^{2} - 24516984 T^{3} + 100167 p^{3} T^{4} - 156 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 122 T + 193581 T^{2} - 17954308 T^{3} + 193581 p^{3} T^{4} - 122 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 504 T + 393661 T^{2} - 109025808 T^{3} + 393661 p^{3} T^{4} - 504 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 282 T + 224563 T^{2} - 87620892 T^{3} + 224563 p^{3} T^{4} - 282 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 548 T + 11419 p T^{2} - 226302104 T^{3} + 11419 p^{4} T^{4} - 548 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 414 T - 389 T^{2} + 154404524 T^{3} - 389 p^{3} T^{4} - 414 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 428 T + 695537 T^{2} + 249317576 T^{3} + 695537 p^{3} T^{4} + 428 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 912 T + 1171621 T^{2} + 649961952 T^{3} + 1171621 p^{3} T^{4} + 912 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 618 T + 1066507 T^{2} - 454366420 T^{3} + 1066507 p^{3} T^{4} - 618 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 542 T + 1317893 T^{2} + 445950836 T^{3} + 1317893 p^{3} T^{4} + 542 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 624021 T^{2} - 434328048 T^{3} + 624021 p^{3} T^{4} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 790 T - 3625 T^{2} + 827778380 T^{3} - 3625 p^{3} T^{4} - 790 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 2074 T + 3705231 T^{2} - 3687692268 T^{3} + 3705231 p^{3} T^{4} - 2074 p^{6} T^{5} + p^{9} T^{6} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.07213620022155376748219150416, −10.95952434991586803381706544673, −10.32905600085776775472269563612, −10.01009874096093577340172841380, −9.268455562508143367854105178668, −9.261540974824869603616116470878, −8.826181451716671637307219185094, −8.550169913998213120309375469828, −8.444290400968193791126627873489, −7.909159536683858479355610257641, −7.61131257479566256921582865126, −7.41371340984739844776743153979, −6.83698751604387676029093115516, −6.56234032380865355909389017182, −6.11635232458765421240275922591, −5.65415127863166200025224192889, −4.45464060830498510112173876351, −4.33647458257485578459268146193, −4.27716707731121188860300728042, −3.55357447941858873412856689763, −3.21416292338059839401969833517, −2.40774915836707764379651066303, −2.07943964802138277745080401681, −1.09047527745476473118115575954, −0.925154523294689183766443694039,
0.925154523294689183766443694039, 1.09047527745476473118115575954, 2.07943964802138277745080401681, 2.40774915836707764379651066303, 3.21416292338059839401969833517, 3.55357447941858873412856689763, 4.27716707731121188860300728042, 4.33647458257485578459268146193, 4.45464060830498510112173876351, 5.65415127863166200025224192889, 6.11635232458765421240275922591, 6.56234032380865355909389017182, 6.83698751604387676029093115516, 7.41371340984739844776743153979, 7.61131257479566256921582865126, 7.909159536683858479355610257641, 8.444290400968193791126627873489, 8.550169913998213120309375469828, 8.826181451716671637307219185094, 9.261540974824869603616116470878, 9.268455562508143367854105178668, 10.01009874096093577340172841380, 10.32905600085776775472269563612, 10.95952434991586803381706544673, 11.07213620022155376748219150416
