| L(s) = 1 | − 3·3-s + 3·5-s − 6·7-s + 6·9-s + 2·11-s + 4·13-s − 9·15-s − 2·17-s − 2·19-s + 18·21-s − 3·23-s + 6·25-s − 10·27-s − 6·29-s − 2·31-s − 6·33-s − 18·35-s − 12·39-s − 2·41-s − 20·43-s + 18·45-s + 6·47-s + 10·49-s + 6·51-s − 6·53-s + 6·55-s + 6·57-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 1.34·5-s − 2.26·7-s + 2·9-s + 0.603·11-s + 1.10·13-s − 2.32·15-s − 0.485·17-s − 0.458·19-s + 3.92·21-s − 0.625·23-s + 6/5·25-s − 1.92·27-s − 1.11·29-s − 0.359·31-s − 1.04·33-s − 3.04·35-s − 1.92·39-s − 0.312·41-s − 3.04·43-s + 2.68·45-s + 0.875·47-s + 10/7·49-s + 0.840·51-s − 0.824·53-s + 0.809·55-s + 0.794·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 5 | $C_1$ | \( ( 1 - T )^{3} \) |
| 23 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 7 | $S_4\times C_2$ | \( 1 + 6 T + 26 T^{2} + 76 T^{3} + 26 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 2 T + 27 T^{2} - 36 T^{3} + 27 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 4 T + 37 T^{2} - 100 T^{3} + 37 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 2 T + 48 T^{2} + 66 T^{3} + 48 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 2 T + 43 T^{2} + 44 T^{3} + 43 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 6 T + 60 T^{2} + 262 T^{3} + 60 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 2 T + 78 T^{2} + 92 T^{3} + 78 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 104 T^{2} - 2 T^{3} + 104 p T^{4} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 2 T - 8 T^{2} - 54 T^{3} - 8 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 20 T + 245 T^{2} + 1896 T^{3} + 245 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 6 T + 119 T^{2} - 580 T^{3} + 119 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 6 T + 132 T^{2} + 582 T^{3} + 132 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 12 T + 134 T^{2} - 1448 T^{3} + 134 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 4 T + 173 T^{2} + 492 T^{3} + 173 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 6 T + 74 T^{2} - 80 T^{3} + 74 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 8 T + 194 T^{2} - 988 T^{3} + 194 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 8 T + 81 T^{2} + 236 T^{3} + 81 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 4 T + 125 T^{2} - 696 T^{3} + 125 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 8 T + 246 T^{2} + 1236 T^{3} + 246 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 6 T + 107 T^{2} - 1084 T^{3} + 107 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 14 T + 131 T^{2} - 540 T^{3} + 131 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.51753910948058578219960947674, −7.01820475239984164932649222759, −6.77567767566924620777290359381, −6.75904527177945357086711005543, −6.50217144066444844475024390964, −6.41620556190591914806967815977, −6.24594280271300422061256946029, −5.85627907693275720042973019102, −5.79559474047823067209460175795, −5.50293879356424681670806400867, −5.12705907743973796983922289746, −5.10690573756739483658473182178, −4.82064496890382895107639353085, −4.22921800295166084921294705549, −4.12953382454728826607417282560, −3.88283372747718658164252006795, −3.41516017393529501970121449245, −3.34695822995481466213377612155, −3.26296651663171467333649252905, −2.39218842680616095565867475805, −2.33061306114660668376013001734, −2.17079265004713863937461432270, −1.38706060858033946867150350593, −1.26363842568701699024378657709, −1.18854558734192236960784281190, 0, 0, 0,
1.18854558734192236960784281190, 1.26363842568701699024378657709, 1.38706060858033946867150350593, 2.17079265004713863937461432270, 2.33061306114660668376013001734, 2.39218842680616095565867475805, 3.26296651663171467333649252905, 3.34695822995481466213377612155, 3.41516017393529501970121449245, 3.88283372747718658164252006795, 4.12953382454728826607417282560, 4.22921800295166084921294705549, 4.82064496890382895107639353085, 5.10690573756739483658473182178, 5.12705907743973796983922289746, 5.50293879356424681670806400867, 5.79559474047823067209460175795, 5.85627907693275720042973019102, 6.24594280271300422061256946029, 6.41620556190591914806967815977, 6.50217144066444844475024390964, 6.75904527177945357086711005543, 6.77567767566924620777290359381, 7.01820475239984164932649222759, 7.51753910948058578219960947674
