| L(s) = 1 | − 2·2-s − 4-s − 3·5-s − 3·7-s + 5·8-s − 2·9-s + 6·10-s − 7·11-s + 5·13-s + 6·14-s − 16-s + 5·17-s + 4·18-s − 6·19-s + 3·20-s + 14·22-s − 3·23-s + 6·25-s − 10·26-s − 7·27-s + 3·28-s − 12·29-s − 4·31-s − 4·32-s − 10·34-s + 9·35-s + 2·36-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1/2·4-s − 1.34·5-s − 1.13·7-s + 1.76·8-s − 2/3·9-s + 1.89·10-s − 2.11·11-s + 1.38·13-s + 1.60·14-s − 1/4·16-s + 1.21·17-s + 0.942·18-s − 1.37·19-s + 0.670·20-s + 2.98·22-s − 0.625·23-s + 6/5·25-s − 1.96·26-s − 1.34·27-s + 0.566·28-s − 2.22·29-s − 0.718·31-s − 0.707·32-s − 1.71·34-s + 1.52·35-s + 1/3·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{3} \cdot 7^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{3} \cdot 7^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 5 | $C_1$ | \( ( 1 + T )^{3} \) |
| 7 | $C_1$ | \( ( 1 + T )^{3} \) |
| 23 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 2 | $A_4\times C_2$ | \( 1 + p T + 5 T^{2} + 7 T^{3} + 5 p T^{4} + p^{3} T^{5} + p^{3} T^{6} \) |
| 3 | $A_4\times C_2$ | \( 1 + 2 T^{2} + 7 T^{3} + 2 p T^{4} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 + 7 T + 47 T^{2} + 161 T^{3} + 47 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $A_4\times C_2$ | \( 1 - 5 T + 17 T^{2} - 33 T^{3} + 17 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 - 5 T + 43 T^{2} - 129 T^{3} + 43 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 + 6 T + 62 T^{2} + 215 T^{3} + 62 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 + 12 T + 107 T^{2} + 592 T^{3} + 107 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 + 4 T + 96 T^{2} + 247 T^{3} + 96 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 - 3 T + 93 T^{2} - 195 T^{3} + 93 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 - T + 58 T^{2} + 87 T^{3} + 58 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 - 2 T + 128 T^{2} - 171 T^{3} + 128 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 - 12 T + 140 T^{2} - 1087 T^{3} + 140 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 + 9 T + 11 T^{2} - 419 T^{3} + 11 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 + 21 T + 303 T^{2} + 2681 T^{3} + 303 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + 9 T + 161 T^{2} + 887 T^{3} + 161 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 - 13 T + 185 T^{2} - 1365 T^{3} + 185 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 + 16 T + 198 T^{2} + 1809 T^{3} + 198 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 + 9 T + 197 T^{2} + 1285 T^{3} + 197 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 18 T + 282 T^{2} + 2493 T^{3} + 282 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 + 29 T + 443 T^{2} + 4603 T^{3} + 443 p T^{4} + 29 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 - 7 T + 253 T^{2} - 1155 T^{3} + 253 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 - T + 142 T^{2} - 277 T^{3} + 142 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
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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
−9.562417468251677658992029589314, −9.110674552816913051647095934735, −8.984847345582920926673361043674, −8.842074896028910377868684174633, −8.424956291227087854705975994943, −8.058470254691835317902668370714, −8.048140721938519622070096694679, −7.59960726974466287279060004096, −7.58493524026161590508768706991, −7.27422672792843348887847050257, −6.81777222133053344278430271644, −6.16210499881621762735684485022, −5.99938800496020904094184202629, −5.66602774536000681594820872108, −5.61007160195289774711931640067, −5.15398351663416817176671487953, −4.47425452011155382749808215230, −4.30294272305617922110948871446, −3.99888896709999327050707342580, −3.66869116963300675576049283818, −3.20990569499169791234434488287, −2.94289741066041825777760458380, −2.54665906620521375314917062626, −1.73288391344918494292290772918, −1.20793493726235163084881198577, 0, 0, 0,
1.20793493726235163084881198577, 1.73288391344918494292290772918, 2.54665906620521375314917062626, 2.94289741066041825777760458380, 3.20990569499169791234434488287, 3.66869116963300675576049283818, 3.99888896709999327050707342580, 4.30294272305617922110948871446, 4.47425452011155382749808215230, 5.15398351663416817176671487953, 5.61007160195289774711931640067, 5.66602774536000681594820872108, 5.99938800496020904094184202629, 6.16210499881621762735684485022, 6.81777222133053344278430271644, 7.27422672792843348887847050257, 7.58493524026161590508768706991, 7.59960726974466287279060004096, 8.048140721938519622070096694679, 8.058470254691835317902668370714, 8.424956291227087854705975994943, 8.842074896028910377868684174633, 8.984847345582920926673361043674, 9.110674552816913051647095934735, 9.562417468251677658992029589314
