| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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
−8.453407752444128035141525862345, −8.322120178589551716424563689467, −8.239012119690227007848983960848, −7.920028212792913808888699565297, −7.56886769887351108433817138122, −7.34763157858977993218573059202, −7.14555099035619371390201214556, −6.54716213425966633017891131726, −6.48013495713356578721358211578, −6.37337248580923672697341728267, −5.50823922266162208765880663122, −5.42541513982929421876956168481, −5.36499607098803742166698835001, −4.83430006546242323052663889032, −4.82754795419624018468690951103, −4.75337085865042495352861238966, −4.01436924759045570102158335123, −3.52751238869622494288749441864, −3.14010192441708057902545619881, −2.91864139660091334910920322923, −2.70947680707328080312257889876, −2.43292877360622137588875974118, −1.79728691094932214265185777940, −1.07888221655774043241416818902, −1.03883609032757913509121252243, 0, 0, 0,
1.03883609032757913509121252243, 1.07888221655774043241416818902, 1.79728691094932214265185777940, 2.43292877360622137588875974118, 2.70947680707328080312257889876, 2.91864139660091334910920322923, 3.14010192441708057902545619881, 3.52751238869622494288749441864, 4.01436924759045570102158335123, 4.75337085865042495352861238966, 4.82754795419624018468690951103, 4.83430006546242323052663889032, 5.36499607098803742166698835001, 5.42541513982929421876956168481, 5.50823922266162208765880663122, 6.37337248580923672697341728267, 6.48013495713356578721358211578, 6.54716213425966633017891131726, 7.14555099035619371390201214556, 7.34763157858977993218573059202, 7.56886769887351108433817138122, 7.920028212792913808888699565297, 8.239012119690227007848983960848, 8.322120178589551716424563689467, 8.453407752444128035141525862345
