| L(s) = 1 | − 2-s − 3·4-s − 14·5-s − 20·7-s − 19·8-s + 14·10-s − 32·13-s + 20·14-s + 49·16-s + 92·17-s + 34·19-s + 42·20-s + 26·23-s + 15·25-s + 32·26-s + 60·28-s + 174·29-s + 422·31-s − 95·32-s − 92·34-s + 280·35-s + 518·37-s − 34·38-s + 266·40-s + 428·41-s + 550·43-s − 26·46-s + ⋯ |
| L(s) = 1 | − 0.353·2-s − 3/8·4-s − 1.25·5-s − 1.07·7-s − 0.839·8-s + 0.442·10-s − 0.682·13-s + 0.381·14-s + 0.765·16-s + 1.31·17-s + 0.410·19-s + 0.469·20-s + 0.235·23-s + 3/25·25-s + 0.241·26-s + 0.404·28-s + 1.11·29-s + 2.44·31-s − 0.524·32-s − 0.464·34-s + 1.35·35-s + 2.30·37-s − 0.145·38-s + 1.05·40-s + 1.63·41-s + 1.95·43-s − 0.0833·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.522284875\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.522284875\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2 \wr S_4$ | \( 1 + T + p^{2} T^{2} + 13 p T^{3} + p^{3} T^{4} + 13 p^{4} T^{5} + p^{8} T^{6} + p^{9} T^{7} + p^{12} T^{8} \) |
| 5 | $C_2 \wr S_4$ | \( 1 + 14 T + 181 T^{2} + 34 T^{3} - 3712 T^{4} + 34 p^{3} T^{5} + 181 p^{6} T^{6} + 14 p^{9} T^{7} + p^{12} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 + 20 T + 745 T^{2} + 250 p T^{3} + 151460 T^{4} + 250 p^{4} T^{5} + 745 p^{6} T^{6} + 20 p^{9} T^{7} + p^{12} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 + 32 T + 6283 T^{2} + 172324 T^{3} + 19678376 T^{4} + 172324 p^{3} T^{5} + 6283 p^{6} T^{6} + 32 p^{9} T^{7} + p^{12} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - 92 T + 10015 T^{2} - 1708 p^{2} T^{3} + 48261836 T^{4} - 1708 p^{5} T^{5} + 10015 p^{6} T^{6} - 92 p^{9} T^{7} + p^{12} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 34 T + 7789 T^{2} - 491934 T^{3} + 100192812 T^{4} - 491934 p^{3} T^{5} + 7789 p^{6} T^{6} - 34 p^{9} T^{7} + p^{12} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - 26 T + 5800 T^{2} - 628762 T^{3} + 272602574 T^{4} - 628762 p^{3} T^{5} + 5800 p^{6} T^{6} - 26 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - 6 p T + 89405 T^{2} - 10079334 T^{3} + 3059530596 T^{4} - 10079334 p^{3} T^{5} + 89405 p^{6} T^{6} - 6 p^{10} T^{7} + p^{12} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 422 T + 121621 T^{2} - 813586 p T^{3} + 4715438036 T^{4} - 813586 p^{4} T^{5} + 121621 p^{6} T^{6} - 422 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 14 p T + 222196 T^{2} - 69969108 T^{3} + 17042241381 T^{4} - 69969108 p^{3} T^{5} + 222196 p^{6} T^{6} - 14 p^{10} T^{7} + p^{12} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 428 T + 281623 T^{2} - 72996448 T^{3} + 27843065288 T^{4} - 72996448 p^{3} T^{5} + 281623 p^{6} T^{6} - 428 p^{9} T^{7} + p^{12} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 550 T + 356860 T^{2} - 122234838 T^{3} + 43628494230 T^{4} - 122234838 p^{3} T^{5} + 356860 p^{6} T^{6} - 550 p^{9} T^{7} + p^{12} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 556 T + 464992 T^{2} + 159161852 T^{3} + 73095550526 T^{4} + 159161852 p^{3} T^{5} + 464992 p^{6} T^{6} + 556 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 882 T + 753641 T^{2} + 356933250 T^{3} + 170508810204 T^{4} + 356933250 p^{3} T^{5} + 753641 p^{6} T^{6} + 882 p^{9} T^{7} + p^{12} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 158 T + 455704 T^{2} + 10644746 T^{3} + 99834013694 T^{4} + 10644746 p^{3} T^{5} + 455704 p^{6} T^{6} - 158 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 290 T + 780649 T^{2} + 160512114 T^{3} + 249052940940 T^{4} + 160512114 p^{3} T^{5} + 780649 p^{6} T^{6} + 290 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 992 T + 1325281 T^{2} - 870459746 T^{3} + 613021090972 T^{4} - 870459746 p^{3} T^{5} + 1325281 p^{6} T^{6} - 992 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 42 T + 203372 T^{2} + 102129390 T^{3} - 39343991994 T^{4} + 102129390 p^{3} T^{5} + 203372 p^{6} T^{6} - 42 p^{9} T^{7} + p^{12} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 + 1274 T + 1771225 T^{2} + 1360918918 T^{3} + 1080945673244 T^{4} + 1360918918 p^{3} T^{5} + 1771225 p^{6} T^{6} + 1274 p^{9} T^{7} + p^{12} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 362 T + 1421725 T^{2} + 271468310 T^{3} + 902647752772 T^{4} + 271468310 p^{3} T^{5} + 1421725 p^{6} T^{6} + 362 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 1500 T + 2217920 T^{2} - 1968382332 T^{3} + 1823815616910 T^{4} - 1968382332 p^{3} T^{5} + 2217920 p^{6} T^{6} - 1500 p^{9} T^{7} + p^{12} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 1428 T + 2335535 T^{2} + 2459628540 T^{3} + 2471453298972 T^{4} + 2459628540 p^{3} T^{5} + 2335535 p^{6} T^{6} + 1428 p^{9} T^{7} + p^{12} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 1052 T + 2038366 T^{2} - 1884713816 T^{3} + 2895539158591 T^{4} - 1884713816 p^{3} T^{5} + 2038366 p^{6} T^{6} - 1052 p^{9} T^{7} + p^{12} T^{8} \) |
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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
−6.58110044744016087915523639491, −6.55326234743991200849023913990, −6.04272867385088389462061170091, −6.00431611458423706674158165220, −5.99693435276591335235300630098, −5.69145837431366490170583048627, −5.23942372061226139658814117480, −5.00452942924226247165611443789, −4.85028194257971671482762130058, −4.43047698686257782677985336286, −4.32796699604168042207193548728, −4.24819002961875699639527408837, −3.84580375131520474737115033758, −3.36269312471279575884719243755, −3.31907029718870667934478898418, −3.08317745921769734507667723604, −2.86909169571047223217791508963, −2.73255167971245204013268332841, −2.35982451555668780958796157596, −1.77938271313265530821529231263, −1.59953783599073944531285469157, −0.801875581609021222861254278699, −0.77698627866952115155431154788, −0.69617978163266418724360735769, −0.23244915958363576208327079456,
0.23244915958363576208327079456, 0.69617978163266418724360735769, 0.77698627866952115155431154788, 0.801875581609021222861254278699, 1.59953783599073944531285469157, 1.77938271313265530821529231263, 2.35982451555668780958796157596, 2.73255167971245204013268332841, 2.86909169571047223217791508963, 3.08317745921769734507667723604, 3.31907029718870667934478898418, 3.36269312471279575884719243755, 3.84580375131520474737115033758, 4.24819002961875699639527408837, 4.32796699604168042207193548728, 4.43047698686257782677985336286, 4.85028194257971671482762130058, 5.00452942924226247165611443789, 5.23942372061226139658814117480, 5.69145837431366490170583048627, 5.99693435276591335235300630098, 6.00431611458423706674158165220, 6.04272867385088389462061170091, 6.55326234743991200849023913990, 6.58110044744016087915523639491
