| L(s) = 1 | + 2-s − 8·4-s − 15·5-s − 24·7-s − 8·8-s − 15·10-s + 70·11-s + 20·13-s − 24·14-s + 16-s − 100·17-s − 57·19-s + 120·20-s + 70·22-s + 56·23-s + 150·25-s + 20·26-s + 192·28-s + 140·29-s − 422·31-s − 7·32-s − 100·34-s + 360·35-s + 140·37-s − 57·38-s + 120·40-s + 130·41-s + ⋯ |
| L(s) = 1 | + 0.353·2-s − 4-s − 1.34·5-s − 1.29·7-s − 0.353·8-s − 0.474·10-s + 1.91·11-s + 0.426·13-s − 0.458·14-s + 1/64·16-s − 1.42·17-s − 0.688·19-s + 1.34·20-s + 0.678·22-s + 0.507·23-s + 6/5·25-s + 0.150·26-s + 1.29·28-s + 0.896·29-s − 2.44·31-s − 0.0386·32-s − 0.504·34-s + 1.73·35-s + 0.622·37-s − 0.243·38-s + 0.474·40-s + 0.495·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p T )^{3} \) |
| 19 | $C_1$ | \( ( 1 + p T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 - T + 9 T^{2} - 9 T^{3} + 9 p^{3} T^{4} - p^{6} T^{5} + p^{9} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 24 T + 1139 T^{2} + 16084 T^{3} + 1139 p^{3} T^{4} + 24 p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 70 T + 4835 T^{2} - 176852 T^{3} + 4835 p^{3} T^{4} - 70 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 20 T + 5853 T^{2} - 73388 T^{3} + 5853 p^{3} T^{4} - 20 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 100 T + 10947 T^{2} + 840456 T^{3} + 10947 p^{3} T^{4} + 100 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 56 T + 17593 T^{2} - 493872 T^{3} + 17593 p^{3} T^{4} - 56 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 140 T + 75885 T^{2} - 6842196 T^{3} + 75885 p^{3} T^{4} - 140 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 422 T + 143289 T^{2} + 27255644 T^{3} + 143289 p^{3} T^{4} + 422 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 140 T + 85789 T^{2} - 7838540 T^{3} + 85789 p^{3} T^{4} - 140 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 130 T + 191013 T^{2} - 15920460 T^{3} + 191013 p^{3} T^{4} - 130 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 286 T + 164595 T^{2} + 24348004 T^{3} + 164595 p^{3} T^{4} + 286 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 792 T + 503249 T^{2} - 178039856 T^{3} + 503249 p^{3} T^{4} - 792 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 434 T + 371431 T^{2} + 126415236 T^{3} + 371431 p^{3} T^{4} + 434 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 478 T + 663817 T^{2} - 194030340 T^{3} + 663817 p^{3} T^{4} - 478 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 650 T + 737535 T^{2} + 294381908 T^{3} + 737535 p^{3} T^{4} + 650 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 1060 T + 1208081 T^{2} + 657646872 T^{3} + 1208081 p^{3} T^{4} + 1060 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 796 T + 1246957 T^{2} + 575726472 T^{3} + 1246957 p^{3} T^{4} + 796 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 218 T + 656551 T^{2} - 33841388 T^{3} + 656551 p^{3} T^{4} - 218 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 1314 T + 1289277 T^{2} + 789056444 T^{3} + 1289277 p^{3} T^{4} + 1314 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 326 T + 10655 p T^{2} + 103390668 T^{3} + 10655 p^{4} T^{4} + 326 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 502 T + 1939413 T^{2} + 707861436 T^{3} + 1939413 p^{3} T^{4} + 502 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 968 T + 1472769 T^{2} - 2040866588 T^{3} + 1472769 p^{3} T^{4} - 968 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−9.221170252323708547241436853610, −8.838251710388559922359644130456, −8.660635412070971518501897665752, −8.483861580723993486093273981592, −7.890728102340781892484261233476, −7.67820923139396924227578617089, −7.23579614311568472240361213329, −7.10357901444265296883012492660, −6.76830283830114143597177815930, −6.44002768763588746075948840897, −6.22788560388344168381363198124, −5.94850693522825607018159331739, −5.65967362772067972446472043144, −4.85402443712847381015375013035, −4.81074398574573681903946592378, −4.55796526101795424970465093244, −4.03867307012425869315325639929, −3.91908660709431843905414852701, −3.79783593194605209821380201097, −3.19678315757022229518468754333, −3.05357319592563533486386149243, −2.50099128529680571385869148110, −1.91303656173200413242620524073, −1.29049795623076368498402229596, −1.07936998446071228773184587362, 0, 0, 0,
1.07936998446071228773184587362, 1.29049795623076368498402229596, 1.91303656173200413242620524073, 2.50099128529680571385869148110, 3.05357319592563533486386149243, 3.19678315757022229518468754333, 3.79783593194605209821380201097, 3.91908660709431843905414852701, 4.03867307012425869315325639929, 4.55796526101795424970465093244, 4.81074398574573681903946592378, 4.85402443712847381015375013035, 5.65967362772067972446472043144, 5.94850693522825607018159331739, 6.22788560388344168381363198124, 6.44002768763588746075948840897, 6.76830283830114143597177815930, 7.10357901444265296883012492660, 7.23579614311568472240361213329, 7.67820923139396924227578617089, 7.890728102340781892484261233476, 8.483861580723993486093273981592, 8.660635412070971518501897665752, 8.838251710388559922359644130456, 9.221170252323708547241436853610
