| L(s) = 1 | − 2-s − 2·5-s − 7-s + 2·10-s + 13-s + 14-s − 16-s − 14·23-s − 3·25-s − 26-s + 4·29-s − 15·31-s − 32-s + 2·35-s + 3·37-s − 4·41-s − 3·43-s + 14·46-s + 18·47-s − 11·49-s + 3·50-s − 6·53-s − 4·58-s + 13·61-s + 15·62-s − 2·64-s − 2·65-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.894·5-s − 0.377·7-s + 0.632·10-s + 0.277·13-s + 0.267·14-s − 1/4·16-s − 2.91·23-s − 3/5·25-s − 0.196·26-s + 0.742·29-s − 2.69·31-s − 0.176·32-s + 0.338·35-s + 0.493·37-s − 0.624·41-s − 0.457·43-s + 2.06·46-s + 2.62·47-s − 1.57·49-s + 0.424·50-s − 0.824·53-s − 0.525·58-s + 1.66·61-s + 1.90·62-s − 1/4·64-s − 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 3 | | \( 1 \) |
| 19 | | \( 1 \) |
| good | 2 | $S_4\times C_2$ | \( 1 + T + T^{2} + T^{3} + p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + 2 T + 7 T^{2} + 8 T^{3} + 7 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + T + 12 T^{2} + 17 T^{3} + 12 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 9 T^{2} + 36 T^{3} + 9 p T^{4} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - T + 18 T^{2} - 29 T^{3} + 18 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 23 | $S_4\times C_2$ | \( 1 + 14 T + 97 T^{2} + 488 T^{3} + 97 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 4 T + 55 T^{2} - 136 T^{3} + 55 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 15 T + 144 T^{2} + 29 p T^{3} + 144 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )^{3} \) |
| 41 | $S_4\times C_2$ | \( 1 + 4 T + 91 T^{2} + 232 T^{3} + 91 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 3 T + 108 T^{2} + 271 T^{3} + 108 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{3} \) |
| 53 | $S_4\times C_2$ | \( 1 + 6 T + 87 T^{2} + 312 T^{3} + 87 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 153 T^{2} + 36 T^{3} + 153 p T^{4} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 13 T + 194 T^{2} - 1513 T^{3} + 194 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 9 T + 120 T^{2} + 665 T^{3} + 120 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 18 T + 225 T^{2} + 1908 T^{3} + 225 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 19 T + 302 T^{2} - 2743 T^{3} + 302 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 11 T + 240 T^{2} + 1567 T^{3} + 240 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 4 T + 169 T^{2} - 472 T^{3} + 169 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 16 T + 343 T^{2} + 2956 T^{3} + 343 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 2 T + 23 T^{2} + 1060 T^{3} + 23 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.076929570686501869880043750905, −7.80592674587353136688785108243, −7.56028957440462094984965561173, −7.49396530746904440763361249367, −7.14686762986946581883910337240, −6.76825454342780826192192577190, −6.59648205045544505347845448050, −6.37859484917464343589040276348, −5.91680556914187330972539485315, −5.80580443649574785977328357614, −5.63437673614617317595133250661, −5.17625257266107517399741821634, −5.11983341559857479201677529066, −4.36097755752949110617355183817, −4.35953655038619654802146625417, −4.12379555688929912367195833355, −3.73038752767358613399884608806, −3.65842970692845583483604765489, −3.27750508057994994716430261207, −2.93352453712820641535816080515, −2.40180154624353660855947826856, −2.19605150690283172513836637949, −1.88862369257073229715430370938, −1.38076250042160071913884739728, −1.10012515852530976888929704079, 0, 0, 0,
1.10012515852530976888929704079, 1.38076250042160071913884739728, 1.88862369257073229715430370938, 2.19605150690283172513836637949, 2.40180154624353660855947826856, 2.93352453712820641535816080515, 3.27750508057994994716430261207, 3.65842970692845583483604765489, 3.73038752767358613399884608806, 4.12379555688929912367195833355, 4.35953655038619654802146625417, 4.36097755752949110617355183817, 5.11983341559857479201677529066, 5.17625257266107517399741821634, 5.63437673614617317595133250661, 5.80580443649574785977328357614, 5.91680556914187330972539485315, 6.37859484917464343589040276348, 6.59648205045544505347845448050, 6.76825454342780826192192577190, 7.14686762986946581883910337240, 7.49396530746904440763361249367, 7.56028957440462094984965561173, 7.80592674587353136688785108243, 8.076929570686501869880043750905
