| L(s) = 1 | + 3·2-s − 3·3-s + 6·4-s + 3·5-s − 9·6-s − 2·7-s + 10·8-s + 6·9-s + 9·10-s − 3·11-s − 18·12-s − 4·13-s − 6·14-s − 9·15-s + 15·16-s − 3·17-s + 18·18-s + 18·20-s + 6·21-s − 9·22-s − 8·23-s − 30·24-s + 6·25-s − 12·26-s − 10·27-s − 12·28-s − 18·29-s + ⋯ |
| L(s) = 1 | + 2.12·2-s − 1.73·3-s + 3·4-s + 1.34·5-s − 3.67·6-s − 0.755·7-s + 3.53·8-s + 2·9-s + 2.84·10-s − 0.904·11-s − 5.19·12-s − 1.10·13-s − 1.60·14-s − 2.32·15-s + 15/4·16-s − 0.727·17-s + 4.24·18-s + 4.02·20-s + 1.30·21-s − 1.91·22-s − 1.66·23-s − 6.12·24-s + 6/5·25-s − 2.35·26-s − 1.92·27-s − 2.26·28-s − 3.34·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 5^{3} \cdot 11^{3} \cdot 17^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 5^{3} \cdot 11^{3} \cdot 17^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 3 | $C_1$ | \( ( 1 + T )^{3} \) |
| 5 | $C_1$ | \( ( 1 - T )^{3} \) |
| 11 | $C_1$ | \( ( 1 + T )^{3} \) |
| 17 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 7 | $A_4\times C_2$ | \( 1 + 2 T + 13 T^{2} + 20 T^{3} + 13 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $A_4\times C_2$ | \( 1 + 4 T + 35 T^{2} + 96 T^{3} + 35 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 + 29 T^{2} - 56 T^{3} + 29 p T^{4} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 + 8 T + 25 T^{2} + 24 T^{3} + 25 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{3} \) |
| 31 | $A_4\times C_2$ | \( 1 - 8 T + 77 T^{2} - 432 T^{3} + 77 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 + 10 T + 107 T^{2} + 636 T^{3} + 107 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{3} \) |
| 43 | $A_4\times C_2$ | \( 1 - 6 T + 113 T^{2} - 412 T^{3} + 113 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 + 29 T^{2} + 448 T^{3} + 29 p T^{4} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 - 2 T + 39 T^{2} - 540 T^{3} + 39 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 + 14 T + 233 T^{2} + 1708 T^{3} + 233 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + 2 T + 147 T^{2} + 236 T^{3} + 147 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 + 2 T + 53 T^{2} + 36 T^{3} + 53 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 + 18 T + 293 T^{2} + 2660 T^{3} + 293 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 + 4 T - 9 T^{2} + 16 T^{3} - 9 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 12 T + 33 T^{2} - 560 T^{3} + 33 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 - 8 T + 121 T^{2} - 816 T^{3} + 121 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 - 2 T + 119 T^{2} - 124 T^{3} + 119 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 + 16 T + 339 T^{2} + 3040 T^{3} + 339 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.44176597581575724761803119268, −6.98501729256649353179269797340, −6.92496162200872508774982734181, −6.77898333050478838991642045346, −6.31760621499586206817978942584, −6.18893646221066901874588070916, −6.17835964747877127025393889412, −5.68393867789381402591988523454, −5.57003102128979362932271299462, −5.50921002293669641943726246051, −5.13299227921604002368354664323, −4.96859087846579295595483920565, −4.75102417951852354061578346953, −4.50201986995816215473238229411, −4.12211771549748166077739048541, −4.07868595263036227307913674290, −3.41119880404180902949418792557, −3.31503344037963716172829841817, −3.27117285363651326273815510075, −2.58353571817856349465894949616, −2.37609768906542248445272480015, −2.20390542785887282223823076639, −1.67729583840738109882228108659, −1.51873037841422847245472957801, −1.39948620824275007486560411553, 0, 0, 0,
1.39948620824275007486560411553, 1.51873037841422847245472957801, 1.67729583840738109882228108659, 2.20390542785887282223823076639, 2.37609768906542248445272480015, 2.58353571817856349465894949616, 3.27117285363651326273815510075, 3.31503344037963716172829841817, 3.41119880404180902949418792557, 4.07868595263036227307913674290, 4.12211771549748166077739048541, 4.50201986995816215473238229411, 4.75102417951852354061578346953, 4.96859087846579295595483920565, 5.13299227921604002368354664323, 5.50921002293669641943726246051, 5.57003102128979362932271299462, 5.68393867789381402591988523454, 6.17835964747877127025393889412, 6.18893646221066901874588070916, 6.31760621499586206817978942584, 6.77898333050478838991642045346, 6.92496162200872508774982734181, 6.98501729256649353179269797340, 7.44176597581575724761803119268
