| L(s) = 1 | + 2-s + 3-s − 3·4-s + 2·5-s + 6-s − 4·8-s − 6·9-s + 2·10-s + 2·11-s − 3·12-s − 13-s + 2·15-s + 3·16-s − 2·17-s − 6·18-s − 13·19-s − 6·20-s + 2·22-s − 3·23-s − 4·24-s − 10·25-s − 26-s − 8·27-s − 18·29-s + 2·30-s + 11·31-s + 6·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 3/2·4-s + 0.894·5-s + 0.408·6-s − 1.41·8-s − 2·9-s + 0.632·10-s + 0.603·11-s − 0.866·12-s − 0.277·13-s + 0.516·15-s + 3/4·16-s − 0.485·17-s − 1.41·18-s − 2.98·19-s − 1.34·20-s + 0.426·22-s − 0.625·23-s − 0.816·24-s − 2·25-s − 0.196·26-s − 1.53·27-s − 3.34·29-s + 0.365·30-s + 1.97·31-s + 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(43^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(43^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 43 | | \( 1 \) |
| good | 2 | $A_4\times C_2$ | \( 1 - T + p^{2} T^{2} - 3 T^{3} + p^{3} T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 3 | $A_4\times C_2$ | \( 1 - T + 7 T^{2} - 5 T^{3} + 7 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $A_4\times C_2$ | \( 1 - 2 T + 14 T^{2} - 19 T^{3} + 14 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 + 2 p T^{2} + p T^{3} + 2 p^{2} T^{4} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 - 2 T + 18 T^{2} - 57 T^{3} + 18 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $A_4\times C_2$ | \( 1 + T + 9 T^{2} + 67 T^{3} + 9 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 + 2 T + 43 T^{2} + 60 T^{3} + 43 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 + 13 T + 111 T^{2} + 565 T^{3} + 111 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 + 3 T + 44 T^{2} + 55 T^{3} + 44 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{3} \) |
| 31 | $A_4\times C_2$ | \( 1 - 11 T + 47 T^{2} - 135 T^{3} + 47 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 + 7 T + 97 T^{2} + 427 T^{3} + 97 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 - 22 T + 275 T^{2} - 52 p T^{3} + 275 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 + 23 T + 301 T^{2} + 2469 T^{3} + 301 p T^{4} + 23 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 - 6 T + 108 T^{2} - 707 T^{3} + 108 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 + 8 T + 182 T^{2} + 943 T^{3} + 182 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 - 6 T + 167 T^{2} - 740 T^{3} + 167 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 - 12 T + 221 T^{2} - 1504 T^{3} + 221 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 + 22 T + 309 T^{2} + 3228 T^{3} + 309 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 + 15 T + 245 T^{2} + 2119 T^{3} + 245 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 10 T + 254 T^{2} + 1581 T^{3} + 254 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 + 4 T + 154 T^{2} + 9 p T^{3} + 154 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 - 2 T + 112 T^{2} - 579 T^{3} + 112 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 + 21 T + 347 T^{2} + 3577 T^{3} + 347 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.800938604878146511455396915692, −8.315689834984677081527451216141, −8.140244197038614680640686366595, −7.972705664119451383008188431846, −7.75830860223088875562133578699, −7.40065460824674825471088425030, −6.83758232481201086190822896629, −6.56851472577406863351431825324, −6.25197616527228533562707993751, −6.09691304797364140761978249465, −5.84648473085253812661475352206, −5.58402338738749320916043821979, −5.54377048876148086820882559033, −4.83687610768424297391441365176, −4.80875649240204605516260383034, −4.28444693337754501419316016558, −4.26139897431680764769833397452, −3.77653547329191098204826637546, −3.71103390656535112823842640209, −3.30497955269179687276884237375, −2.67648950962309843253572876528, −2.57744811110109640540126886464, −2.16673502630378030672172129931, −1.78070726273135151543339422458, −1.49080644779524810862631328236, 0, 0, 0,
1.49080644779524810862631328236, 1.78070726273135151543339422458, 2.16673502630378030672172129931, 2.57744811110109640540126886464, 2.67648950962309843253572876528, 3.30497955269179687276884237375, 3.71103390656535112823842640209, 3.77653547329191098204826637546, 4.26139897431680764769833397452, 4.28444693337754501419316016558, 4.80875649240204605516260383034, 4.83687610768424297391441365176, 5.54377048876148086820882559033, 5.58402338738749320916043821979, 5.84648473085253812661475352206, 6.09691304797364140761978249465, 6.25197616527228533562707993751, 6.56851472577406863351431825324, 6.83758232481201086190822896629, 7.40065460824674825471088425030, 7.75830860223088875562133578699, 7.972705664119451383008188431846, 8.140244197038614680640686366595, 8.315689834984677081527451216141, 8.800938604878146511455396915692
