| L(s) = 1 | − 15·5-s − 27·11-s − 3·13-s + 15·17-s + 78·19-s − 105·23-s + 150·25-s + 117·29-s + 207·31-s − 120·37-s + 300·41-s + 483·43-s − 303·47-s − 522·49-s − 492·53-s + 405·55-s − 240·59-s − 444·61-s + 45·65-s + 522·67-s + 168·71-s − 876·73-s + 2.10e3·79-s + 42·83-s − 225·85-s − 2.26e3·89-s − 1.17e3·95-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 0.740·11-s − 0.0640·13-s + 0.214·17-s + 0.941·19-s − 0.951·23-s + 6/5·25-s + 0.749·29-s + 1.19·31-s − 0.533·37-s + 1.14·41-s + 1.71·43-s − 0.940·47-s − 1.52·49-s − 1.27·53-s + 0.992·55-s − 0.529·59-s − 0.931·61-s + 0.0858·65-s + 0.951·67-s + 0.280·71-s − 1.40·73-s + 2.99·79-s + 0.0555·83-s − 0.287·85-s − 2.70·89-s − 1.26·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{9} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{9} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p T )^{3} \) |
| good | 7 | $S_4\times C_2$ | \( 1 + 522 T^{2} + 2666 T^{3} + 522 p^{3} T^{4} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 27 T + 2721 T^{2} + 79918 T^{3} + 2721 p^{3} T^{4} + 27 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 3 T + 4962 T^{2} + 32735 T^{3} + 4962 p^{3} T^{4} + 3 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 15 T + 5031 T^{2} + 273298 T^{3} + 5031 p^{3} T^{4} - 15 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 78 T + 9942 T^{2} - 303500 T^{3} + 9942 p^{3} T^{4} - 78 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 105 T + 25413 T^{2} + 2052802 T^{3} + 25413 p^{3} T^{4} + 105 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 117 T + 67683 T^{2} - 5431966 T^{3} + 67683 p^{3} T^{4} - 117 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 207 T + 69741 T^{2} - 8616098 T^{3} + 69741 p^{3} T^{4} - 207 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 120 T - 39264 T^{2} - 10421922 T^{3} - 39264 p^{3} T^{4} + 120 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 300 T + 82023 T^{2} - 31686600 T^{3} + 82023 p^{3} T^{4} - 300 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 483 T + 251961 T^{2} - 76432898 T^{3} + 251961 p^{3} T^{4} - 483 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 303 T + 166137 T^{2} + 65064754 T^{3} + 166137 p^{3} T^{4} + 303 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 492 T + 476259 T^{2} + 138372072 T^{3} + 476259 p^{3} T^{4} + 492 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 240 T + 213705 T^{2} - 18543136 T^{3} + 213705 p^{3} T^{4} + 240 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 444 T + 185052 T^{2} - 860902 T^{3} + 185052 p^{3} T^{4} + 444 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 522 T + 370530 T^{2} - 307525264 T^{3} + 370530 p^{3} T^{4} - 522 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 168 T - 41151 T^{2} + 401172816 T^{3} - 41151 p^{3} T^{4} - 168 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 12 p T + 1148628 T^{2} + 668051142 T^{3} + 1148628 p^{3} T^{4} + 12 p^{7} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 2103 T + 2885448 T^{2} - 2370776439 T^{3} + 2885448 p^{3} T^{4} - 2103 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 42 T + 653901 T^{2} - 416514772 T^{3} + 653901 p^{3} T^{4} - 42 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 2268 T + 3125643 T^{2} + 3038479992 T^{3} + 3125643 p^{3} T^{4} + 2268 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 1392 T + 3023340 T^{2} + 2460493802 T^{3} + 3023340 p^{3} T^{4} + 1392 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−8.137064890879103645521573549043, −7.83104772430791490291724644342, −7.79817856160809709654317473298, −7.37374695200219440911758462995, −6.97213289207322803188708023747, −6.79123496512286260423989110319, −6.75902127479819184164013205585, −6.04046729726759971709142345410, −5.95516631835115663484458385839, −5.94027397320684799618793145298, −5.22753459566887183435608429972, −5.09082201110169866802286346054, −4.87216745054287742950958759868, −4.53070112805184184698084242312, −4.17959256754443170408852974307, −4.11855572336999968744912447344, −3.56219126542119659374365313754, −3.30107397215558925859693112215, −3.18335535799564069755117718311, −2.59306058190371649028922223128, −2.50783497274334256762745621734, −2.16964427624555545296542056583, −1.33755958000519068152290664465, −1.15193194002268645502200313976, −1.04842071290843893553575578125, 0, 0, 0,
1.04842071290843893553575578125, 1.15193194002268645502200313976, 1.33755958000519068152290664465, 2.16964427624555545296542056583, 2.50783497274334256762745621734, 2.59306058190371649028922223128, 3.18335535799564069755117718311, 3.30107397215558925859693112215, 3.56219126542119659374365313754, 4.11855572336999968744912447344, 4.17959256754443170408852974307, 4.53070112805184184698084242312, 4.87216745054287742950958759868, 5.09082201110169866802286346054, 5.22753459566887183435608429972, 5.94027397320684799618793145298, 5.95516631835115663484458385839, 6.04046729726759971709142345410, 6.75902127479819184164013205585, 6.79123496512286260423989110319, 6.97213289207322803188708023747, 7.37374695200219440911758462995, 7.79817856160809709654317473298, 7.83104772430791490291724644342, 8.137064890879103645521573549043
