| L(s) = 1 | − 3·2-s + 2·3-s + 6·4-s − 2·5-s − 6·6-s + 3·7-s − 10·8-s − 9-s + 6·10-s + 3·11-s + 12·12-s − 5·13-s − 9·14-s − 4·15-s + 15·16-s − 3·17-s + 3·18-s − 9·19-s − 12·20-s + 6·21-s − 9·22-s + 4·23-s − 20·24-s − 7·25-s + 15·26-s − 7·27-s + 18·28-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 1.15·3-s + 3·4-s − 0.894·5-s − 2.44·6-s + 1.13·7-s − 3.53·8-s − 1/3·9-s + 1.89·10-s + 0.904·11-s + 3.46·12-s − 1.38·13-s − 2.40·14-s − 1.03·15-s + 15/4·16-s − 0.727·17-s + 0.707·18-s − 2.06·19-s − 2.68·20-s + 1.30·21-s − 1.91·22-s + 0.834·23-s − 4.08·24-s − 7/5·25-s + 2.94·26-s − 1.34·27-s + 3.40·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 17^{3} \cdot 59^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 17^{3} \cdot 59^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 17 | $C_1$ | \( ( 1 + T )^{3} \) |
| 59 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 - 2 T + 5 T^{2} - 5 T^{3} + 5 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + 2 T + 11 T^{2} + 13 T^{3} + 11 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 3 T + 8 T^{2} - 19 T^{3} + 8 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 3 T + 32 T^{2} - 64 T^{3} + 32 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 5 T + 32 T^{2} + 132 T^{3} + 32 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 9 T + 68 T^{2} + 313 T^{3} + 68 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 4 T + 49 T^{2} - 168 T^{3} + 49 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 13 T + 111 T^{2} + 642 T^{3} + 111 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 12 T + 98 T^{2} + 530 T^{3} + 98 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 13 T + 140 T^{2} + 870 T^{3} + 140 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 7 T + 89 T^{2} + 338 T^{3} + 89 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 16 T + 202 T^{2} + 1450 T^{3} + 202 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 10 T + 124 T^{2} - 678 T^{3} + 124 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 2 T + 23 T^{2} - 199 T^{3} + 23 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 2 T + 64 T^{2} + 572 T^{3} + 64 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 2 T + 152 T^{2} + 366 T^{3} + 152 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 32 T + 548 T^{2} - 5692 T^{3} + 548 p T^{4} - 32 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 6 T + 188 T^{2} + 904 T^{3} + 188 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 12 T + 3 p T^{2} + 1667 T^{3} + 3 p^{2} T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 3 T + 248 T^{2} + 496 T^{3} + 248 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 14 T + 320 T^{2} + 2548 T^{3} + 320 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 15 T + 266 T^{2} - 2410 T^{3} + 266 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.609637737227395361535592952287, −8.154404551357667837589339753750, −8.114631263401174734808471033490, −8.003808492659287874551253956546, −7.51978444568993133418312033040, −7.46443410939105471935052411441, −7.01756280276192793457543068048, −6.86216162765568926989756008481, −6.80161921701301168165688601957, −6.38978151522755775923262378650, −5.81878945203092263424813749467, −5.67226573043648165482404113016, −5.32701542180818372760275926575, −5.04150096926204498538260831968, −4.75520213460815440838379890815, −4.12723455269208642688416994283, −3.84878406652379894506263852138, −3.64546001760203116821425795972, −3.57892688404309531029195081841, −2.91829760947042258089367122732, −2.43250262314289215746985003930, −2.32409806579437273135988957733, −1.84373881041159627909457825934, −1.67374709714653977088972412275, −1.47052493605156283858682293210, 0, 0, 0,
1.47052493605156283858682293210, 1.67374709714653977088972412275, 1.84373881041159627909457825934, 2.32409806579437273135988957733, 2.43250262314289215746985003930, 2.91829760947042258089367122732, 3.57892688404309531029195081841, 3.64546001760203116821425795972, 3.84878406652379894506263852138, 4.12723455269208642688416994283, 4.75520213460815440838379890815, 5.04150096926204498538260831968, 5.32701542180818372760275926575, 5.67226573043648165482404113016, 5.81878945203092263424813749467, 6.38978151522755775923262378650, 6.80161921701301168165688601957, 6.86216162765568926989756008481, 7.01756280276192793457543068048, 7.46443410939105471935052411441, 7.51978444568993133418312033040, 8.003808492659287874551253956546, 8.114631263401174734808471033490, 8.154404551357667837589339753750, 8.609637737227395361535592952287
