| L(s) = 1 | + 3·2-s − 3·3-s + 6·4-s + 3·5-s − 9·6-s + 10·8-s + 6·9-s + 9·10-s − 2·11-s − 18·12-s − 9·15-s + 15·16-s + 3·17-s + 18·18-s + 5·19-s + 18·20-s − 6·22-s + 3·23-s − 30·24-s + 6·25-s − 10·27-s − 10·29-s − 27·30-s + 4·31-s + 21·32-s + 6·33-s + 9·34-s + ⋯ |
| L(s) = 1 | + 2.12·2-s − 1.73·3-s + 3·4-s + 1.34·5-s − 3.67·6-s + 3.53·8-s + 2·9-s + 2.84·10-s − 0.603·11-s − 5.19·12-s − 2.32·15-s + 15/4·16-s + 0.727·17-s + 4.24·18-s + 1.14·19-s + 4.02·20-s − 1.27·22-s + 0.625·23-s − 6.12·24-s + 6/5·25-s − 1.92·27-s − 1.85·29-s − 4.92·30-s + 0.718·31-s + 3.71·32-s + 1.04·33-s + 1.54·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 5^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 5^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(22.27706535\) |
| \(L(\frac12)\) |
\(\approx\) |
\(22.27706535\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 13 | | \( 1 \) |
| good | 7 | $A_4\times C_2$ | \( 1 + 2 p T^{2} - p T^{3} + 2 p^{2} T^{4} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 + 2 T + 18 T^{2} + 57 T^{3} + 18 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 - 3 T + 47 T^{2} - 89 T^{3} + 47 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 - 5 T + 63 T^{2} - 191 T^{3} + 63 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 - 3 T + 51 T^{2} - 111 T^{3} + 51 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 + 10 T + 104 T^{2} + 539 T^{3} + 104 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 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 | $A_4\times C_2$ | \( 1 + 2 T + 110 T^{2} + 147 T^{3} + 110 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 - 17 T + 175 T^{2} - 1227 T^{3} + 175 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 - 19 T + 219 T^{2} - 1747 T^{3} + 219 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 - 7 T + 92 T^{2} - 371 T^{3} + 92 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 - 19 T + 221 T^{2} - 1931 T^{3} + 221 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 - 19 T + 253 T^{2} - 2313 T^{3} + 253 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 - T + 97 T^{2} - 373 T^{3} + 97 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 + T + 129 T^{2} - 35 T^{3} + 129 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 + T + 99 T^{2} - 279 T^{3} + 99 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 + 15 T + 77 T^{2} + 47 T^{3} + 77 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 - 15 T + 291 T^{2} - 2383 T^{3} + 291 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 - 6 T + 212 T^{2} - 857 T^{3} + 212 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 - 9 T + 147 T^{2} - 1825 T^{3} + 147 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 + 4 T + 280 T^{2} + 775 T^{3} + 280 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.29345850354620783381956193739, −6.73594369238836365707173734377, −6.73417003395807437439405756773, −6.52699384724217175233130323014, −6.03185376408312085062821192200, −5.75995452705764168383936045021, −5.67846028066303987411594358665, −5.56530387514592173957019746677, −5.46101666609990206396471951987, −5.39280135009957791750937130881, −4.68379685580992027078147465904, −4.67126656860582450042478400239, −4.52938804735462543343049329762, −3.92327484355580957450920794546, −3.84398177600325721796342233746, −3.81195017545511677045912165084, −2.97378071450217817959047333144, −2.90433027097196402728531525410, −2.80445705041291362075024464616, −2.18118109815535274438921371647, −1.97906144632198144712803635959, −1.83540998482046399607611568812, −1.05949674760602724186523704552, −0.852685078120170273589080533590, −0.72585712268704062390585265377,
0.72585712268704062390585265377, 0.852685078120170273589080533590, 1.05949674760602724186523704552, 1.83540998482046399607611568812, 1.97906144632198144712803635959, 2.18118109815535274438921371647, 2.80445705041291362075024464616, 2.90433027097196402728531525410, 2.97378071450217817959047333144, 3.81195017545511677045912165084, 3.84398177600325721796342233746, 3.92327484355580957450920794546, 4.52938804735462543343049329762, 4.67126656860582450042478400239, 4.68379685580992027078147465904, 5.39280135009957791750937130881, 5.46101666609990206396471951987, 5.56530387514592173957019746677, 5.67846028066303987411594358665, 5.75995452705764168383936045021, 6.03185376408312085062821192200, 6.52699384724217175233130323014, 6.73417003395807437439405756773, 6.73594369238836365707173734377, 7.29345850354620783381956193739
