| L(s) = 1 | − 2-s − 6·4-s + 15·5-s − 43·7-s − 15·10-s + 14·11-s + 40·13-s + 43·14-s − 16-s − 166·17-s − 164·19-s − 90·20-s − 14·22-s + 171·23-s + 150·25-s − 40·26-s + 258·28-s − 335·29-s − 352·31-s + 71·32-s + 166·34-s − 645·35-s + 402·37-s + 164·38-s + 187·41-s − 602·43-s − 84·44-s + ⋯ |
| L(s) = 1 | − 0.353·2-s − 3/4·4-s + 1.34·5-s − 2.32·7-s − 0.474·10-s + 0.383·11-s + 0.853·13-s + 0.820·14-s − 0.0156·16-s − 2.36·17-s − 1.98·19-s − 1.00·20-s − 0.135·22-s + 1.55·23-s + 6/5·25-s − 0.301·26-s + 1.74·28-s − 2.14·29-s − 2.03·31-s + 0.392·32-s + 0.837·34-s − 3.11·35-s + 1.78·37-s + 0.700·38-s + 0.712·41-s − 2.13·43-s − 0.287·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 66430125 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66430125 ^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - p T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 + T + 7 T^{2} + 13 T^{3} + 7 p^{3} T^{4} + p^{6} T^{5} + p^{9} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 43 T + 1224 T^{2} + 24161 T^{3} + 1224 p^{3} T^{4} + 43 p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 14 T + 107 p T^{2} - 80816 T^{3} + 107 p^{4} T^{4} - 14 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 40 T + 4139 T^{2} - 100396 T^{3} + 4139 p^{3} T^{4} - 40 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 166 T + 23659 T^{2} + 1787440 T^{3} + 23659 p^{3} T^{4} + 166 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 164 T + 27869 T^{2} + 2307068 T^{3} + 27869 p^{3} T^{4} + 164 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 171 T + 31668 T^{2} - 4099905 T^{3} + 31668 p^{3} T^{4} - 171 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 335 T + 100498 T^{2} + 16233563 T^{3} + 100498 p^{3} T^{4} + 335 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 352 T + 75441 T^{2} + 11111924 T^{3} + 75441 p^{3} T^{4} + 352 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 402 T + 176667 T^{2} - 37389728 T^{3} + 176667 p^{3} T^{4} - 402 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 187 T + 162166 T^{2} - 18548179 T^{3} + 162166 p^{3} T^{4} - 187 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 14 p T + 248477 T^{2} + 80281904 T^{3} + 248477 p^{3} T^{4} + 14 p^{7} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 665 T + 406750 T^{2} - 140572073 T^{3} + 406750 p^{3} T^{4} - 665 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 730 T + 552931 T^{2} + 220610956 T^{3} + 552931 p^{3} T^{4} + 730 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 298 T + 261493 T^{2} - 4970012 T^{3} + 261493 p^{3} T^{4} + 298 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 1439 T + 1270250 T^{2} + 708864815 T^{3} + 1270250 p^{3} T^{4} + 1439 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 1849 T + 1995966 T^{2} + 1320997733 T^{3} + 1995966 p^{3} T^{4} + 1849 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 70 T + 388273 T^{2} + 173667512 T^{3} + 388273 p^{3} T^{4} - 70 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 368 T + 794123 T^{2} + 151388768 T^{3} + 794123 p^{3} T^{4} + 368 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 382 T + 1091193 T^{2} + 238359212 T^{3} + 1091193 p^{3} T^{4} + 382 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 831 T + 554388 T^{2} - 5533827 T^{3} + 554388 p^{3} T^{4} + 831 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 1719 T + 2353398 T^{2} - 2298177027 T^{3} + 2353398 p^{3} T^{4} - 1719 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 282 T + 2670351 T^{2} + 497849308 T^{3} + 2670351 p^{3} T^{4} + 282 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−9.983998711619038447682764097610, −9.427229710000842326459043743265, −9.353104879237615558517271498693, −9.263915550416091765521365810651, −9.007920393603724184827094486240, −8.766238457940642009707002295898, −8.705064356640272361350817850249, −7.78198917260965230225193949356, −7.66107425279408472166484796289, −7.08981863602694559404947191349, −6.74806795507664424535872186321, −6.51936857598590904718316523056, −6.15351137774435862970517732417, −6.14154487269679994398155530644, −5.74661470241413914151714312229, −5.30095883628344485014299751185, −4.63155593026877467019211178082, −4.28921387377713224539709325720, −4.25785469816814921017030590132, −3.43674415870720298916800428285, −3.26282067515033001679207004643, −2.70539323450455699494528846291, −2.35049074009932263747777905854, −1.65679112050516327814094621851, −1.40419267616158714528906795252, 0, 0, 0,
1.40419267616158714528906795252, 1.65679112050516327814094621851, 2.35049074009932263747777905854, 2.70539323450455699494528846291, 3.26282067515033001679207004643, 3.43674415870720298916800428285, 4.25785469816814921017030590132, 4.28921387377713224539709325720, 4.63155593026877467019211178082, 5.30095883628344485014299751185, 5.74661470241413914151714312229, 6.14154487269679994398155530644, 6.15351137774435862970517732417, 6.51936857598590904718316523056, 6.74806795507664424535872186321, 7.08981863602694559404947191349, 7.66107425279408472166484796289, 7.78198917260965230225193949356, 8.705064356640272361350817850249, 8.766238457940642009707002295898, 9.007920393603724184827094486240, 9.263915550416091765521365810651, 9.353104879237615558517271498693, 9.427229710000842326459043743265, 9.983998711619038447682764097610
