| L(s) = 1 | + 5·2-s + 9·4-s + 15·5-s − 4·7-s + 15·8-s + 75·10-s + 5·11-s + 7·13-s − 20·14-s + 53·16-s + 155·17-s − 50·19-s + 135·20-s + 25·22-s + 285·23-s + 150·25-s + 35·26-s − 36·28-s + 115·29-s − 115·31-s + 155·32-s + 775·34-s − 60·35-s − 384·37-s − 250·38-s + 225·40-s + 580·41-s + ⋯ |
| L(s) = 1 | + 1.76·2-s + 9/8·4-s + 1.34·5-s − 0.215·7-s + 0.662·8-s + 2.37·10-s + 0.137·11-s + 0.149·13-s − 0.381·14-s + 0.828·16-s + 2.21·17-s − 0.603·19-s + 1.50·20-s + 0.242·22-s + 2.58·23-s + 6/5·25-s + 0.264·26-s − 0.242·28-s + 0.736·29-s − 0.666·31-s + 0.856·32-s + 3.90·34-s − 0.289·35-s − 1.70·37-s − 1.06·38-s + 0.889·40-s + 2.20·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2460375 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2460375 ^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(11.10781704\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.10781704\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - p T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 - 5 T + p^{4} T^{2} - 25 p T^{3} + p^{7} T^{4} - 5 p^{6} T^{5} + p^{9} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 4 T + 234 T^{2} - 5554 T^{3} + 234 p^{3} T^{4} + 4 p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 5 T + 1105 T^{2} + 17950 T^{3} + 1105 p^{3} T^{4} - 5 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 7 T + 5822 T^{2} - 37183 T^{3} + 5822 p^{3} T^{4} - 7 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 155 T + 20347 T^{2} - 1564790 T^{3} + 20347 p^{3} T^{4} - 155 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 50 T + 206 p T^{2} + 317888 T^{3} + 206 p^{4} T^{4} + 50 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 285 T + 52701 T^{2} - 6381690 T^{3} + 52701 p^{3} T^{4} - 285 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 115 T + 25759 T^{2} + 830870 T^{3} + 25759 p^{3} T^{4} - 115 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 115 T + 60141 T^{2} + 5913626 T^{3} + 60141 p^{3} T^{4} + 115 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 384 T + 84036 T^{2} + 16234306 T^{3} + 84036 p^{3} T^{4} + 384 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 580 T + 296575 T^{2} - 83865640 T^{3} + 296575 p^{3} T^{4} - 580 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 797 T + 381041 T^{2} + 121376222 T^{3} + 381041 p^{3} T^{4} + 797 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 145 T + 62317 T^{2} + 44496910 T^{3} + 62317 p^{3} T^{4} + 145 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 400 T + 388459 T^{2} + 106443280 T^{3} + 388459 p^{3} T^{4} + 400 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 380 T + 588745 T^{2} - 150882920 T^{3} + 588745 p^{3} T^{4} - 380 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 152 T + 593468 T^{2} + 63933158 T^{3} + 593468 p^{3} T^{4} + 152 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 2 T + 658290 T^{2} + 19566248 T^{3} + 658290 p^{3} T^{4} - 2 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 40 T + 396361 T^{2} + 187438400 T^{3} + 396361 p^{3} T^{4} - 40 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 980 T + 1431584 T^{2} + 778921274 T^{3} + 1431584 p^{3} T^{4} + 980 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 1013 T + 1531332 T^{2} - 908300089 T^{3} + 1531332 p^{3} T^{4} - 1013 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 270 T + 1413933 T^{2} - 224225820 T^{3} + 1413933 p^{3} T^{4} - 270 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 1020 T + 2161707 T^{2} - 1313072760 T^{3} + 2161707 p^{3} T^{4} - 1020 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 720 T + 2703456 T^{2} - 1286818562 T^{3} + 2703456 p^{3} T^{4} - 720 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−11.60092514191548874754994905469, −10.96913094984489451978254615327, −10.86501632263758945479301866937, −10.25061735782117591736534269600, −10.21901100310839782682038709612, −9.640763673825080410958123124545, −9.486032794316793730076526843229, −8.839837742895997453616951799801, −8.654723645012659809167143009417, −8.171529742128186629101289393691, −7.49354775321479957545988309214, −7.30648946861892812463628468798, −6.74403561770058056568235677406, −6.39714317555827353831646835322, −5.90100708404861763999410979545, −5.69352480564299846496836785901, −4.88656520652513150134616365760, −4.86622130777510368577436397152, −4.85833407247916650396246333140, −3.61926941319441371162782051718, −3.46000291396205748825899277840, −3.07916926871191843506904711816, −2.26401802971586434376665794269, −1.51615447186264782629634075631, −0.928288195279676937966404069461,
0.928288195279676937966404069461, 1.51615447186264782629634075631, 2.26401802971586434376665794269, 3.07916926871191843506904711816, 3.46000291396205748825899277840, 3.61926941319441371162782051718, 4.85833407247916650396246333140, 4.86622130777510368577436397152, 4.88656520652513150134616365760, 5.69352480564299846496836785901, 5.90100708404861763999410979545, 6.39714317555827353831646835322, 6.74403561770058056568235677406, 7.30648946861892812463628468798, 7.49354775321479957545988309214, 8.171529742128186629101289393691, 8.654723645012659809167143009417, 8.839837742895997453616951799801, 9.486032794316793730076526843229, 9.640763673825080410958123124545, 10.21901100310839782682038709612, 10.25061735782117591736534269600, 10.86501632263758945479301866937, 10.96913094984489451978254615327, 11.60092514191548874754994905469
