| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{9} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{9} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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
−8.955017829682490303976794759255, −8.484760335152097939183262111949, −8.240000244353871589437763552647, −8.131224180690229464624917330075, −7.61014114423984114972701984111, −7.37045127062988763165137841770, −7.31627565495259460617140764297, −6.69527127558629306174959252850, −6.66661620627109873543208028839, −6.50895665599774887214397179947, −5.91841561752859858523204519052, −5.62289566389347645539522540398, −5.40826933796911370235317525215, −4.80343809108110059144737602975, −4.79787542421900943883923591816, −4.28198195320684409275493171054, −4.05631004826485067995347827926, −3.85424453797724763450498610233, −3.40606131883157535961881430756, −3.02591750431436536117035434708, −2.65133832926645505051318710832, −2.54203006092817928821594324137, −1.47578136262274374908855908286, −1.38151848846479159611088245231, −1.26785452361844842505032605540, 0, 0, 0,
1.26785452361844842505032605540, 1.38151848846479159611088245231, 1.47578136262274374908855908286, 2.54203006092817928821594324137, 2.65133832926645505051318710832, 3.02591750431436536117035434708, 3.40606131883157535961881430756, 3.85424453797724763450498610233, 4.05631004826485067995347827926, 4.28198195320684409275493171054, 4.79787542421900943883923591816, 4.80343809108110059144737602975, 5.40826933796911370235317525215, 5.62289566389347645539522540398, 5.91841561752859858523204519052, 6.50895665599774887214397179947, 6.66661620627109873543208028839, 6.69527127558629306174959252850, 7.31627565495259460617140764297, 7.37045127062988763165137841770, 7.61014114423984114972701984111, 8.131224180690229464624917330075, 8.240000244353871589437763552647, 8.484760335152097939183262111949, 8.955017829682490303976794759255
