| L(s) = 1 | + 4·3-s − 4·5-s + 4·7-s + 10·9-s − 4·11-s − 2·13-s − 16·15-s − 2·17-s − 6·19-s + 16·21-s − 2·23-s + 10·25-s + 20·27-s + 2·31-s − 16·33-s − 16·35-s − 14·37-s − 8·39-s − 2·41-s − 4·43-s − 40·45-s − 10·47-s + 10·49-s − 8·51-s − 22·53-s + 16·55-s − 24·57-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 1.78·5-s + 1.51·7-s + 10/3·9-s − 1.20·11-s − 0.554·13-s − 4.13·15-s − 0.485·17-s − 1.37·19-s + 3.49·21-s − 0.417·23-s + 2·25-s + 3.84·27-s + 0.359·31-s − 2.78·33-s − 2.70·35-s − 2.30·37-s − 1.28·39-s − 0.312·41-s − 0.609·43-s − 5.96·45-s − 1.45·47-s + 10/7·49-s − 1.12·51-s − 3.02·53-s + 2.15·55-s − 3.17·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{4} \) |
| 5 | $C_1$ | \( ( 1 + T )^{4} \) |
| 7 | $C_1$ | \( ( 1 - T )^{4} \) |
| 11 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 13 | $C_2 \wr S_4$ | \( 1 + 2 T + 34 T^{2} + 46 T^{3} + 586 T^{4} + 46 p T^{5} + 34 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 2 T + 33 T^{2} + 82 T^{3} + 660 T^{4} + 82 p T^{5} + 33 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 6 T + 53 T^{2} + 230 T^{3} + 1308 T^{4} + 230 p T^{5} + 53 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + 2 T + 57 T^{2} + 210 T^{3} + 1532 T^{4} + 210 p T^{5} + 57 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + 13 T^{2} - 12 T^{3} + 1636 T^{4} - 12 p T^{5} + 13 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 2 T + 38 T^{2} + 70 T^{3} + 594 T^{4} + 70 p T^{5} + 38 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 + 14 T + 202 T^{2} + 1602 T^{3} + 12218 T^{4} + 1602 p T^{5} + 202 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 2 T + 106 T^{2} - 26 T^{3} + 5050 T^{4} - 26 p T^{5} + 106 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 4 T + 41 T^{2} - 148 T^{3} + 292 T^{4} - 148 p T^{5} + 41 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 10 T + 102 T^{2} + 882 T^{3} + 5618 T^{4} + 882 p T^{5} + 102 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 22 T + 257 T^{2} + 2598 T^{3} + 22124 T^{4} + 2598 p T^{5} + 257 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 16 T + 301 T^{2} + 2912 T^{3} + 28572 T^{4} + 2912 p T^{5} + 301 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 6 T + 221 T^{2} + 894 T^{3} + 19204 T^{4} + 894 p T^{5} + 221 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 12 T + 176 T^{2} + 1516 T^{3} + 14862 T^{4} + 1516 p T^{5} + 176 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 2 T + 230 T^{2} + 394 T^{3} + 23218 T^{4} + 394 p T^{5} + 230 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 8 T + 208 T^{2} - 1896 T^{3} + 19742 T^{4} - 1896 p T^{5} + 208 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 2 T + 134 T^{2} - 442 T^{3} + 15634 T^{4} - 442 p T^{5} + 134 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + 10 T + 121 T^{2} - 22 T^{3} + 2020 T^{4} - 22 p T^{5} + 121 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 4 T + 345 T^{2} + 1016 T^{3} + 45540 T^{4} + 1016 p T^{5} + 345 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 4 T + p T^{2} + 192 T^{3} + 5972 T^{4} + 192 p T^{5} + p^{3} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.97462815987795671052236135432, −5.31114374957970639207951556458, −5.23452314926950735457747988910, −4.96999950759834094958990115467, −4.95758303389098950537517784876, −4.75385004489708165964679245513, −4.74278033224822264642548672939, −4.46213421745924742700851600003, −4.37791519837250788989197574472, −3.91153641726428256862010518725, −3.85369339399679279465587240083, −3.77917745544200554992216159543, −3.68339126117293741879564846782, −3.14588250606290808171993367558, −3.13324018292815261679630024795, −2.99985767197166320232243146023, −2.93764055795603929928375583585, −2.33014898706957958876203962485, −2.30497129353258663470973598126, −2.19925255976059210506683555173, −2.14676686086541189810073935977, −1.39441075172413787915461537228, −1.36157635477794120176427619075, −1.31901603777653894760738478293, −1.31565375221734365209350573474, 0, 0, 0, 0,
1.31565375221734365209350573474, 1.31901603777653894760738478293, 1.36157635477794120176427619075, 1.39441075172413787915461537228, 2.14676686086541189810073935977, 2.19925255976059210506683555173, 2.30497129353258663470973598126, 2.33014898706957958876203962485, 2.93764055795603929928375583585, 2.99985767197166320232243146023, 3.13324018292815261679630024795, 3.14588250606290808171993367558, 3.68339126117293741879564846782, 3.77917745544200554992216159543, 3.85369339399679279465587240083, 3.91153641726428256862010518725, 4.37791519837250788989197574472, 4.46213421745924742700851600003, 4.74278033224822264642548672939, 4.75385004489708165964679245513, 4.95758303389098950537517784876, 4.96999950759834094958990115467, 5.23452314926950735457747988910, 5.31114374957970639207951556458, 5.97462815987795671052236135432
