| L(s) = 1 | − 5·2-s + 7·4-s − 8·5-s − 8·7-s + 9·8-s + 40·10-s − 34·11-s + 36·13-s + 40·14-s − 85·16-s − 51·17-s − 142·19-s − 56·20-s + 170·22-s − 110·23-s − 252·25-s − 180·26-s − 56·28-s − 90·29-s − 148·31-s + 341·32-s + 255·34-s + 64·35-s + 110·37-s + 710·38-s − 72·40-s − 720·41-s + ⋯ |
| L(s) = 1 | − 1.76·2-s + 7/8·4-s − 0.715·5-s − 0.431·7-s + 0.397·8-s + 1.26·10-s − 0.931·11-s + 0.768·13-s + 0.763·14-s − 1.32·16-s − 0.727·17-s − 1.71·19-s − 0.626·20-s + 1.64·22-s − 0.997·23-s − 2.01·25-s − 1.35·26-s − 0.377·28-s − 0.576·29-s − 0.857·31-s + 1.88·32-s + 1.28·34-s + 0.309·35-s + 0.488·37-s + 3.03·38-s − 0.284·40-s − 2.74·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3581577 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3581577 ^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 17 | $C_1$ | \( ( 1 + p T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 + 5 T + 9 p T^{2} + 23 p T^{3} + 9 p^{4} T^{4} + 5 p^{6} T^{5} + p^{9} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + 8 T + 316 T^{2} + 1534 T^{3} + 316 p^{3} T^{4} + 8 p^{6} T^{5} + p^{9} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 8 T + 51 p T^{2} + 7792 T^{3} + 51 p^{4} T^{4} + 8 p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 34 T + 2450 T^{2} + 99472 T^{3} + 2450 p^{3} T^{4} + 34 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 36 T + 1060 T^{2} - 35486 T^{3} + 1060 p^{3} T^{4} - 36 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 142 T + 24690 T^{2} + 1956200 T^{3} + 24690 p^{3} T^{4} + 142 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 110 T + 30814 T^{2} + 2729980 T^{3} + 30814 p^{3} T^{4} + 110 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 90 T + 36139 T^{2} + 4805340 T^{3} + 36139 p^{3} T^{4} + 90 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 148 T + 82401 T^{2} + 8177688 T^{3} + 82401 p^{3} T^{4} + 148 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 110 T + 71031 T^{2} - 17113452 T^{3} + 71031 p^{3} T^{4} - 110 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 720 T + 366208 T^{2} + 109686282 T^{3} + 366208 p^{3} T^{4} + 720 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 146 T - 40182 T^{2} - 39408872 T^{3} - 40182 p^{3} T^{4} + 146 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 500 T + 217553 T^{2} + 73350104 T^{3} + 217553 p^{3} T^{4} + 500 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 610 T + 340931 T^{2} + 101182252 T^{3} + 340931 p^{3} T^{4} + 610 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 216 T + 576349 T^{2} - 90026112 T^{3} + 576349 p^{3} T^{4} - 216 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 18 T + 539791 T^{2} + 16298516 T^{3} + 539791 p^{3} T^{4} + 18 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 1404 T + 1477377 T^{2} + 906611816 T^{3} + 1477377 p^{3} T^{4} + 1404 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 960 T + 856597 T^{2} - 459564544 T^{3} + 856597 p^{3} T^{4} - 960 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 794 T + 908231 T^{2} + 390276652 T^{3} + 908231 p^{3} T^{4} + 794 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 276 T + 332913 T^{2} + 51343320 T^{3} + 332913 p^{3} T^{4} + 276 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 1552 T + 2256501 T^{2} - 1786088240 T^{3} + 2256501 p^{3} T^{4} - 1552 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 1394 T + 2215963 T^{2} + 1686994660 T^{3} + 2215963 p^{3} T^{4} + 1394 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 402 T + 119587 T^{2} + 1292278940 T^{3} + 119587 p^{3} T^{4} - 402 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−11.65644661908976076971321866031, −11.21163196919064666561215266294, −10.96965997318068539077218433579, −10.50400401685843674619413223268, −10.16420266429647573781039250093, −9.838401726963965250614617301965, −9.778701373800632889397996087190, −9.068366168463588903802152142660, −8.897996537578487955617633066410, −8.644243741732146534086964003420, −8.142182592671096268013463040669, −7.88229179255241623027027399994, −7.83474739392216022566928054928, −7.22694930000157491336706397957, −6.56917714299354251397802095120, −6.32754042281170777383415357055, −6.17627826640342648198481797484, −5.17720788170549705638842089984, −5.08004569547810011465154599049, −4.14971281380827279224524226351, −4.08902997194734833753093184761, −3.47203228415881083087051326614, −2.77838577401269894070988374506, −1.80764492040638991603309607579, −1.75646561566549658193007977758, 0, 0, 0,
1.75646561566549658193007977758, 1.80764492040638991603309607579, 2.77838577401269894070988374506, 3.47203228415881083087051326614, 4.08902997194734833753093184761, 4.14971281380827279224524226351, 5.08004569547810011465154599049, 5.17720788170549705638842089984, 6.17627826640342648198481797484, 6.32754042281170777383415357055, 6.56917714299354251397802095120, 7.22694930000157491336706397957, 7.83474739392216022566928054928, 7.88229179255241623027027399994, 8.142182592671096268013463040669, 8.644243741732146534086964003420, 8.897996537578487955617633066410, 9.068366168463588903802152142660, 9.778701373800632889397996087190, 9.838401726963965250614617301965, 10.16420266429647573781039250093, 10.50400401685843674619413223268, 10.96965997318068539077218433579, 11.21163196919064666561215266294, 11.65644661908976076971321866031
