| L(s) = 1 | − 4·2-s − 11·3-s − 7·4-s − 27·5-s + 44·6-s − 20·7-s + 38·8-s + 2·9-s + 108·10-s − 62·11-s + 77·12-s − 2·13-s + 80·14-s + 297·15-s + 95·16-s − 207·17-s − 8·18-s + 99·19-s + 189·20-s + 220·21-s + 248·22-s − 103·23-s − 418·24-s + 64·25-s + 8·26-s + 448·27-s + 140·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 2.11·3-s − 7/8·4-s − 2.41·5-s + 2.99·6-s − 1.07·7-s + 1.67·8-s + 2/27·9-s + 3.41·10-s − 1.69·11-s + 1.85·12-s − 0.0426·13-s + 1.52·14-s + 5.11·15-s + 1.48·16-s − 2.95·17-s − 0.104·18-s + 1.19·19-s + 2.11·20-s + 2.28·21-s + 2.40·22-s − 0.933·23-s − 3.55·24-s + 0.511·25-s + 0.0603·26-s + 3.19·27-s + 0.944·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3418801 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3418801 ^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 43 | $C_1$ | \( ( 1 - p T )^{4} \) |
| good | 2 | $C_2 \wr S_4$ | \( 1 + p^{2} T + 23 T^{2} + 41 p T^{3} + 121 p T^{4} + 41 p^{4} T^{5} + 23 p^{6} T^{6} + p^{11} T^{7} + p^{12} T^{8} \) |
| 3 | $C_2 \wr S_4$ | \( 1 + 11 T + 119 T^{2} + 839 T^{3} + 4996 T^{4} + 839 p^{3} T^{5} + 119 p^{6} T^{6} + 11 p^{9} T^{7} + p^{12} T^{8} \) |
| 5 | $C_2 \wr S_4$ | \( 1 + 27 T + 133 p T^{2} + 9849 T^{3} + 134272 T^{4} + 9849 p^{3} T^{5} + 133 p^{7} T^{6} + 27 p^{9} T^{7} + p^{12} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 + 20 T + 768 T^{2} + 13108 T^{3} + 314942 T^{4} + 13108 p^{3} T^{5} + 768 p^{6} T^{6} + 20 p^{9} T^{7} + p^{12} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 62 T + 3077 T^{2} + 39962 T^{3} + 1317884 T^{4} + 39962 p^{3} T^{5} + 3077 p^{6} T^{6} + 62 p^{9} T^{7} + p^{12} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 + 2 T + 8273 T^{2} + 102 p^{2} T^{3} + 26690692 T^{4} + 102 p^{5} T^{5} + 8273 p^{6} T^{6} + 2 p^{9} T^{7} + p^{12} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 207 T + 14258 T^{2} - 190032 T^{3} - 71042783 T^{4} - 190032 p^{3} T^{5} + 14258 p^{6} T^{6} + 207 p^{9} T^{7} + p^{12} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 99 T + 22201 T^{2} - 1662075 T^{3} + 222917608 T^{4} - 1662075 p^{3} T^{5} + 22201 p^{6} T^{6} - 99 p^{9} T^{7} + p^{12} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + 103 T + 1466 p T^{2} + 2793222 T^{3} + 586593953 T^{4} + 2793222 p^{3} T^{5} + 1466 p^{7} T^{6} + 103 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + 99 T - 13567 T^{2} - 167967 T^{3} + 888213804 T^{4} - 167967 p^{3} T^{5} - 13567 p^{6} T^{6} + 99 p^{9} T^{7} + p^{12} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 131 T + 64436 T^{2} - 3696474 T^{3} + 1950279047 T^{4} - 3696474 p^{3} T^{5} + 64436 p^{6} T^{6} - 131 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 + 449 T + 266817 T^{2} + 71392239 T^{3} + 21942685768 T^{4} + 71392239 p^{3} T^{5} + 266817 p^{6} T^{6} + 449 p^{9} T^{7} + p^{12} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 491 T + 187430 T^{2} + 55413200 T^{3} + 16109516117 T^{4} + 55413200 p^{3} T^{5} + 187430 p^{6} T^{6} + 491 p^{9} T^{7} + p^{12} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 - 19 T + 108577 T^{2} - 15073647 T^{3} + 11126438636 T^{4} - 15073647 p^{3} T^{5} + 108577 p^{6} T^{6} - 19 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 1220 T + 1002413 T^{2} + 551938550 T^{3} + 246387730196 T^{4} + 551938550 p^{3} T^{5} + 1002413 p^{6} T^{6} + 1220 p^{9} T^{7} + p^{12} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 816 T + 987944 T^{2} - 513131184 T^{3} + 321570111646 T^{4} - 513131184 p^{3} T^{5} + 987944 p^{6} T^{6} - 816 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 372 T + 843468 T^{2} - 226528412 T^{3} + 278949483798 T^{4} - 226528412 p^{3} T^{5} + 843468 p^{6} T^{6} - 372 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 110 T + 855641 T^{2} - 172379298 T^{3} + 331907899304 T^{4} - 172379298 p^{3} T^{5} + 855641 p^{6} T^{6} - 110 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 468 T + 925804 T^{2} - 494793108 T^{3} + 407494264950 T^{4} - 494793108 p^{3} T^{5} + 925804 p^{6} T^{6} - 468 p^{9} T^{7} + p^{12} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 628 T + 1127772 T^{2} - 544172556 T^{3} + 627880103782 T^{4} - 544172556 p^{3} T^{5} + 1127772 p^{6} T^{6} - 628 p^{9} T^{7} + p^{12} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 1095 T + 1663851 T^{2} - 1399726691 T^{3} + 1195813590120 T^{4} - 1399726691 p^{3} T^{5} + 1663851 p^{6} T^{6} - 1095 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + 980 T + 1568765 T^{2} + 705601262 T^{3} + 896947637720 T^{4} + 705601262 p^{3} T^{5} + 1568765 p^{6} T^{6} + 980 p^{9} T^{7} + p^{12} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 738 T + 1788008 T^{2} + 1031145174 T^{3} + 1620376904526 T^{4} + 1031145174 p^{3} T^{5} + 1788008 p^{6} T^{6} + 738 p^{9} T^{7} + p^{12} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + 1765 T + 3796458 T^{2} + 4179032168 T^{3} + 5180997869897 T^{4} + 4179032168 p^{3} T^{5} + 3796458 p^{6} T^{6} + 1765 p^{9} T^{7} + p^{12} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.01780983287906378542144311656, −11.48838420658883003803899633792, −11.42230515320839694342187025958, −11.21804509606757029592750993271, −10.88475641635290898768089814608, −10.71342009696540652481881570164, −9.907665259651408127799374816789, −9.794178876615938390308309220110, −9.676807659860829215719399668753, −9.092568994018558002623118507769, −8.683131246248817719857982159307, −8.396601300798307261886086510184, −8.171147250820017242419173390129, −7.948877530633172318215697462441, −7.64575871593026355654384151066, −6.77873077730597500092482137380, −6.68027283943148003200848791882, −6.15602159798918147310067970245, −5.58051308712182402002748960698, −5.29201696417632416344218292323, −5.04730727682708990420924376260, −4.42670586904208046656720741517, −3.72297841755081739373342928538, −3.56361457069440822121808531120, −2.65296324650834500624809691174, 0, 0, 0, 0,
2.65296324650834500624809691174, 3.56361457069440822121808531120, 3.72297841755081739373342928538, 4.42670586904208046656720741517, 5.04730727682708990420924376260, 5.29201696417632416344218292323, 5.58051308712182402002748960698, 6.15602159798918147310067970245, 6.68027283943148003200848791882, 6.77873077730597500092482137380, 7.64575871593026355654384151066, 7.948877530633172318215697462441, 8.171147250820017242419173390129, 8.396601300798307261886086510184, 8.683131246248817719857982159307, 9.092568994018558002623118507769, 9.676807659860829215719399668753, 9.794178876615938390308309220110, 9.907665259651408127799374816789, 10.71342009696540652481881570164, 10.88475641635290898768089814608, 11.21804509606757029592750993271, 11.42230515320839694342187025958, 11.48838420658883003803899633792, 12.01780983287906378542144311656
