| L(s) = 1 | − 3·3-s − 3·7-s + 6·9-s − 2·11-s + 2·13-s − 2·19-s + 9·21-s − 8·23-s − 10·27-s + 2·29-s + 2·31-s + 6·33-s − 4·37-s − 6·39-s + 14·41-s − 12·43-s − 8·47-s + 6·49-s + 14·53-s + 6·57-s − 8·59-s + 2·61-s − 18·63-s + 24·69-s + 6·71-s + 6·73-s + 6·77-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 1.13·7-s + 2·9-s − 0.603·11-s + 0.554·13-s − 0.458·19-s + 1.96·21-s − 1.66·23-s − 1.92·27-s + 0.371·29-s + 0.359·31-s + 1.04·33-s − 0.657·37-s − 0.960·39-s + 2.18·41-s − 1.82·43-s − 1.16·47-s + 6/7·49-s + 1.92·53-s + 0.794·57-s − 1.04·59-s + 0.256·61-s − 2.26·63-s + 2.88·69-s + 0.712·71-s + 0.702·73-s + 0.683·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{6} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{6} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.640514697\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.640514697\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 11 | $S_4\times C_2$ | \( 1 + 2 T + 13 T^{2} + 36 T^{3} + 13 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $D_{6}$ | \( 1 - 2 T + 27 T^{2} - 44 T^{3} + 27 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 35 T^{2} + 16 T^{3} + 35 p T^{4} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 2 T + 45 T^{2} + 68 T^{3} + 45 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 8 T + 77 T^{2} + 352 T^{3} + 77 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $D_{6}$ | \( 1 - 2 T + 3 T^{2} - 12 T^{3} + 3 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 2 T + 81 T^{2} - 116 T^{3} + 81 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $D_{6}$ | \( 1 + 4 T + 63 T^{2} + 232 T^{3} + 63 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 14 T + 175 T^{2} - 1188 T^{3} + 175 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 12 T + 113 T^{2} + 712 T^{3} + 113 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 8 T + 77 T^{2} + 496 T^{3} + 77 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 14 T + 131 T^{2} - 1012 T^{3} + 131 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 8 T + 113 T^{2} + 688 T^{3} + 113 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 2 T + 35 T^{2} - 780 T^{3} + 35 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 71 | $S_4\times C_2$ | \( 1 - 6 T + 113 T^{2} - 1052 T^{3} + 113 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 6 T + 95 T^{2} - 116 T^{3} + 95 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 20 T + 317 T^{2} - 3096 T^{3} + 317 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 8 T + 185 T^{2} + 1072 T^{3} + 185 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 2 T + 207 T^{2} - 156 T^{3} + 207 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 14 T + 295 T^{2} + 2372 T^{3} + 295 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−6.90961827225270672235154543117, −6.52326360575162948041090584901, −6.33750164649226312968202019361, −6.28964707019974493132793197468, −6.01026306473603639513835167261, −5.70917907657490054648972227718, −5.55776470657225303675916823168, −5.23348397665148064140240375067, −5.19456160569620445575579451299, −4.87571883055870957007638076748, −4.35543007256786164787287386825, −4.35297890463641706636489022088, −4.18274900224986497455678749070, −3.64086493167821816867331059024, −3.61902846868471272167498135726, −3.45319531266720259585057520139, −2.73985390657052978550879214822, −2.63808955219945409002013083213, −2.58528366037530987558107251950, −1.77737058994670304910461619279, −1.64812499181677481084137011284, −1.64093586109539227995507544300, −0.63375706179281654840435651085, −0.57708639316787773147104037929, −0.45276716010462051671441067729,
0.45276716010462051671441067729, 0.57708639316787773147104037929, 0.63375706179281654840435651085, 1.64093586109539227995507544300, 1.64812499181677481084137011284, 1.77737058994670304910461619279, 2.58528366037530987558107251950, 2.63808955219945409002013083213, 2.73985390657052978550879214822, 3.45319531266720259585057520139, 3.61902846868471272167498135726, 3.64086493167821816867331059024, 4.18274900224986497455678749070, 4.35297890463641706636489022088, 4.35543007256786164787287386825, 4.87571883055870957007638076748, 5.19456160569620445575579451299, 5.23348397665148064140240375067, 5.55776470657225303675916823168, 5.70917907657490054648972227718, 6.01026306473603639513835167261, 6.28964707019974493132793197468, 6.33750164649226312968202019361, 6.52326360575162948041090584901, 6.90961827225270672235154543117
