| L(s) = 1 | − 3·2-s − 3·3-s + 6·4-s + 3·5-s + 9·6-s + 3·7-s − 10·8-s + 6·9-s − 9·10-s − 4·11-s − 18·12-s + 6·13-s − 9·14-s − 9·15-s + 15·16-s − 2·17-s − 18·18-s + 4·19-s + 18·20-s − 9·21-s + 12·22-s + 3·23-s + 30·24-s + 6·25-s − 18·26-s − 10·27-s + 18·28-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.73·3-s + 3·4-s + 1.34·5-s + 3.67·6-s + 1.13·7-s − 3.53·8-s + 2·9-s − 2.84·10-s − 1.20·11-s − 5.19·12-s + 1.66·13-s − 2.40·14-s − 2.32·15-s + 15/4·16-s − 0.485·17-s − 4.24·18-s + 0.917·19-s + 4.02·20-s − 1.96·21-s + 2.55·22-s + 0.625·23-s + 6.12·24-s + 6/5·25-s − 3.53·26-s − 1.92·27-s + 3.40·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 5^{3} \cdot 7^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 5^{3} \cdot 7^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.906871942\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.906871942\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 7 | $C_1$ | \( ( 1 - T )^{3} \) |
| 23 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 11 | $S_4\times C_2$ | \( 1 + 4 T + 17 T^{2} + 56 T^{3} + 17 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) |
| 17 | $S_4\times C_2$ | \( 1 + 2 T + 15 T^{2} - 36 T^{3} + 15 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 4 T + 41 T^{2} - 120 T^{3} + 41 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 2 T + 67 T^{2} - 108 T^{3} + 67 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 8 T + 77 T^{2} + 336 T^{3} + 77 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 10 T + 91 T^{2} - 604 T^{3} + 91 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 2 T + 71 T^{2} - 124 T^{3} + 71 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 8 T + 113 T^{2} - 528 T^{3} + 113 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 4 T + 125 T^{2} + 344 T^{3} + 125 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 10 T + 107 T^{2} - 860 T^{3} + 107 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 12 T + 161 T^{2} + 1096 T^{3} + 161 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{3} \) |
| 67 | $S_4\times C_2$ | \( 1 - 8 T + 89 T^{2} - 144 T^{3} + 89 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 8 T + 101 T^{2} + 208 T^{3} + 101 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 14 T + 199 T^{2} - 1700 T^{3} + 199 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 8 T + 205 T^{2} - 1136 T^{3} + 205 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 8 T + 89 T^{2} - 784 T^{3} + 89 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 6 T + 215 T^{2} - 1076 T^{3} + 215 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 26 T + 463 T^{2} - 5228 T^{3} + 463 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.50122139305648892105078515590, −6.94471496941925186034845927028, −6.89310955964678388548611015378, −6.75775601485698006089589886019, −6.25729836144522353770867097716, −6.09259002521498271133503064106, −6.07348126546478389549866794875, −5.57465920257156988028942161781, −5.50288641163873385637802384711, −5.37687229787142460247644789936, −4.90074790696534751558911227187, −4.80400219014163381435567943115, −4.53979234923654555512357177624, −3.87679803790337598087955834275, −3.66095526434883254499225195081, −3.63227232509339132469976146955, −2.73145681318141231636148453790, −2.58934832775269338105112379198, −2.57959559208125832895087879224, −1.82064994541473295953762419728, −1.68230379597080096388294680186, −1.60517022328019728030378973693, −0.803946905097636973156489406291, −0.78292712091054739650768518648, −0.60942129493842629452153743254,
0.60942129493842629452153743254, 0.78292712091054739650768518648, 0.803946905097636973156489406291, 1.60517022328019728030378973693, 1.68230379597080096388294680186, 1.82064994541473295953762419728, 2.57959559208125832895087879224, 2.58934832775269338105112379198, 2.73145681318141231636148453790, 3.63227232509339132469976146955, 3.66095526434883254499225195081, 3.87679803790337598087955834275, 4.53979234923654555512357177624, 4.80400219014163381435567943115, 4.90074790696534751558911227187, 5.37687229787142460247644789936, 5.50288641163873385637802384711, 5.57465920257156988028942161781, 6.07348126546478389549866794875, 6.09259002521498271133503064106, 6.25729836144522353770867097716, 6.75775601485698006089589886019, 6.89310955964678388548611015378, 6.94471496941925186034845927028, 7.50122139305648892105078515590
