| L(s) = 1 | − 2·2-s + 3·3-s − 4-s − 6·6-s + 3·7-s + 5·8-s + 6·9-s − 2·11-s − 3·12-s − 6·14-s − 16-s − 2·17-s − 12·18-s − 13·19-s + 9·21-s + 4·22-s + 3·23-s + 15·24-s − 8·25-s + 10·27-s − 3·28-s + 29-s − 11·31-s − 4·32-s − 6·33-s + 4·34-s − 6·36-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.73·3-s − 1/2·4-s − 2.44·6-s + 1.13·7-s + 1.76·8-s + 2·9-s − 0.603·11-s − 0.866·12-s − 1.60·14-s − 1/4·16-s − 0.485·17-s − 2.82·18-s − 2.98·19-s + 1.96·21-s + 0.852·22-s + 0.625·23-s + 3.06·24-s − 8/5·25-s + 1.92·27-s − 0.566·28-s + 0.185·29-s − 1.97·31-s − 0.707·32-s − 1.04·33-s + 0.685·34-s − 36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 7^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 7^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 7 | $C_1$ | \( ( 1 - T )^{3} \) |
| 13 | | \( 1 \) |
| good | 2 | $A_4\times C_2$ | \( 1 + p T + 5 T^{2} + 7 T^{3} + 5 p T^{4} + p^{3} T^{5} + p^{3} T^{6} \) |
| 5 | $A_4\times C_2$ | \( 1 + 8 T^{2} + 7 T^{3} + 8 p T^{4} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 + 2 T + 18 T^{2} + 57 T^{3} + 18 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 + 2 T + 50 T^{2} + 67 T^{3} + 50 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 + 13 T + 104 T^{2} + 537 T^{3} + 104 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 - 3 T + p T^{2} + T^{3} + p^{2} T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 - T + 15 T^{2} + 111 T^{3} + 15 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 + 11 T + 89 T^{2} + 471 T^{3} + 89 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 + 4 T + 2 T^{2} - 237 T^{3} + 2 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 - 2 T + 87 T^{2} - 156 T^{3} + 87 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 + 11 T + 125 T^{2} + 735 T^{3} + 125 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 + 19 T + 245 T^{2} + 1913 T^{3} + 245 p T^{4} + 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 - 5 T + 130 T^{2} - 489 T^{3} + 130 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 + 11 T + 173 T^{2} + 1087 T^{3} + 173 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 - 7 T + 78 T^{2} - 175 T^{3} + 78 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 + 15 T + 227 T^{2} + 1981 T^{3} + 227 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 + 18 T + 230 T^{2} + 2023 T^{3} + 230 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 + 6 T + 140 T^{2} + 499 T^{3} + 140 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 - 3 T + 219 T^{2} - 447 T^{3} + 219 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 + 2 T + 150 T^{2} + 345 T^{3} + 150 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 + 17 T + 354 T^{2} + 3153 T^{3} + 354 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 + 13 T + 261 T^{2} + 2509 T^{3} + 261 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.162214425160112791331466433775, −7.79593077639991618329221457996, −7.68019281564045708146452295825, −7.65185198078703459013652866119, −6.93937348707371645391840774910, −6.88515125167483096005396497278, −6.77296910947666258576074852982, −6.33381892144647774200893288278, −5.85561970018706551447892867893, −5.79734658788605406975393929460, −5.24387575691723931803626402578, −5.17471286392607298525133074449, −4.70550230437079965476378681394, −4.43809654484409714396579303484, −4.41235093677462099126956384983, −4.04264921375194176530977180528, −3.79019196008661210808140997214, −3.52527349002934946727276506956, −3.00898382061728924538700438566, −2.84530957505091452253674933720, −2.30748928525852596279724052650, −2.19047891621209693559846132728, −1.61910398369233951363610760759, −1.43661095772659531419589913364, −1.42694668003177434549560651178, 0, 0, 0,
1.42694668003177434549560651178, 1.43661095772659531419589913364, 1.61910398369233951363610760759, 2.19047891621209693559846132728, 2.30748928525852596279724052650, 2.84530957505091452253674933720, 3.00898382061728924538700438566, 3.52527349002934946727276506956, 3.79019196008661210808140997214, 4.04264921375194176530977180528, 4.41235093677462099126956384983, 4.43809654484409714396579303484, 4.70550230437079965476378681394, 5.17471286392607298525133074449, 5.24387575691723931803626402578, 5.79734658788605406975393929460, 5.85561970018706551447892867893, 6.33381892144647774200893288278, 6.77296910947666258576074852982, 6.88515125167483096005396497278, 6.93937348707371645391840774910, 7.65185198078703459013652866119, 7.68019281564045708146452295825, 7.79593077639991618329221457996, 8.162214425160112791331466433775
