| L(s) = 1 | − 4·3-s + 7·5-s − 7·7-s − 22·9-s − 103·11-s + 32·13-s − 28·15-s + 11·17-s + 57·19-s + 28·21-s − 316·23-s − 82·25-s + 86·27-s − 138·29-s − 420·31-s + 412·33-s − 49·35-s + 102·37-s − 128·39-s − 370·41-s − 431·43-s − 154·45-s + 199·47-s − 891·49-s − 44·51-s − 308·53-s − 721·55-s + ⋯ |
| L(s) = 1 | − 0.769·3-s + 0.626·5-s − 0.377·7-s − 0.814·9-s − 2.82·11-s + 0.682·13-s − 0.481·15-s + 0.156·17-s + 0.688·19-s + 0.290·21-s − 2.86·23-s − 0.655·25-s + 0.612·27-s − 0.883·29-s − 2.43·31-s + 2.17·33-s − 0.236·35-s + 0.453·37-s − 0.525·39-s − 1.40·41-s − 1.52·43-s − 0.510·45-s + 0.617·47-s − 2.59·49-s − 0.120·51-s − 0.798·53-s − 1.76·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28094464 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28094464 ^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - p T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + 4 T + 38 T^{2} + 154 T^{3} + 38 p^{3} T^{4} + 4 p^{6} T^{5} + p^{9} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - 7 T + 131 T^{2} - 1142 T^{3} + 131 p^{3} T^{4} - 7 p^{6} T^{5} + p^{9} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + p T + 940 T^{2} + 4243 T^{3} + 940 p^{3} T^{4} + p^{7} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 103 T + 6205 T^{2} + 252030 T^{3} + 6205 p^{3} T^{4} + 103 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 32 T + 2914 T^{2} - 9188 T^{3} + 2914 p^{3} T^{4} - 32 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 11 T + 666 p T^{2} - 140259 T^{3} + 666 p^{4} T^{4} - 11 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 316 T + 59648 T^{2} + 7447208 T^{3} + 59648 p^{3} T^{4} + 316 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 138 T + 73586 T^{2} + 6385598 T^{3} + 73586 p^{3} T^{4} + 138 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 420 T + 144957 T^{2} + 27360696 T^{3} + 144957 p^{3} T^{4} + 420 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 102 T - 10893 T^{2} + 5232188 T^{3} - 10893 p^{3} T^{4} - 102 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 370 T + 242923 T^{2} + 51708996 T^{3} + 242923 p^{3} T^{4} + 370 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 431 T + 218013 T^{2} + 63114554 T^{3} + 218013 p^{3} T^{4} + 431 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 199 T + 96357 T^{2} + 3984558 T^{3} + 96357 p^{3} T^{4} - 199 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 308 T + 105730 T^{2} + 98338872 T^{3} + 105730 p^{3} T^{4} + 308 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 188 T + 613942 T^{2} - 76145262 T^{3} + 613942 p^{3} T^{4} - 188 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 609 T + 637811 T^{2} + 225844310 T^{3} + 637811 p^{3} T^{4} + 609 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 246 T + 507842 T^{2} - 51728764 T^{3} + 507842 p^{3} T^{4} - 246 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 954 T + 904961 T^{2} - 445620740 T^{3} + 904961 p^{3} T^{4} - 954 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 629 T + 522206 T^{2} + 72331857 T^{3} + 522206 p^{3} T^{4} + 629 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 452 T + 451225 T^{2} - 155904568 T^{3} + 451225 p^{3} T^{4} + 452 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 780 T + 311213 T^{2} + 156799240 T^{3} + 311213 p^{3} T^{4} - 780 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 1356 T + 2683643 T^{2} + 1983548248 T^{3} + 2683643 p^{3} T^{4} + 1356 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 548 T + 149095 T^{2} - 1035193144 T^{3} + 149095 p^{3} T^{4} + 548 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−10.26763580195074676103040040632, −10.22162387804147810646705125220, −10.07631081111263262342216007006, −9.795715569884797775139934354644, −9.145943323624515733377138700284, −9.132516529612352791773007412671, −8.507810117694480635837482060761, −8.079308047283427314572612299458, −7.938278985992462597987326589712, −7.72927760168752107329528784030, −7.41846989957823225875675103310, −6.70556912393122874192339818158, −6.48096883962091282516400275652, −5.96353221531555567172647901773, −5.73667459731354845032724034809, −5.55399477507143820531959854976, −5.25541532288174959219124090638, −4.92511699806790761660551946803, −4.35366855168567833371880230420, −3.63276152953702285859411546318, −3.39450308257032131912545305695, −3.04479141633118990360676554759, −2.18539184943696806942614263097, −2.08914211250791010900706832071, −1.49090226129081519846287847615, 0, 0, 0,
1.49090226129081519846287847615, 2.08914211250791010900706832071, 2.18539184943696806942614263097, 3.04479141633118990360676554759, 3.39450308257032131912545305695, 3.63276152953702285859411546318, 4.35366855168567833371880230420, 4.92511699806790761660551946803, 5.25541532288174959219124090638, 5.55399477507143820531959854976, 5.73667459731354845032724034809, 5.96353221531555567172647901773, 6.48096883962091282516400275652, 6.70556912393122874192339818158, 7.41846989957823225875675103310, 7.72927760168752107329528784030, 7.938278985992462597987326589712, 8.079308047283427314572612299458, 8.507810117694480635837482060761, 9.132516529612352791773007412671, 9.145943323624515733377138700284, 9.795715569884797775139934354644, 10.07631081111263262342216007006, 10.22162387804147810646705125220, 10.26763580195074676103040040632
