| L(s) = 1 | + 2-s + 3-s − 4-s − 6·5-s + 6-s − 2·8-s − 6·10-s − 6·11-s − 12-s − 9·13-s − 6·15-s − 3·16-s − 17-s − 13·19-s + 6·20-s − 6·22-s − 11·23-s − 2·24-s + 9·25-s − 9·26-s + 2·27-s + 10·29-s − 6·30-s + 2·31-s − 6·33-s − 34-s + 3·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s − 2.68·5-s + 0.408·6-s − 0.707·8-s − 1.89·10-s − 1.80·11-s − 0.288·12-s − 2.49·13-s − 1.54·15-s − 3/4·16-s − 0.242·17-s − 2.98·19-s + 1.34·20-s − 1.27·22-s − 2.29·23-s − 0.408·24-s + 9/5·25-s − 1.76·26-s + 0.384·27-s + 1.85·29-s − 1.09·30-s + 0.359·31-s − 1.04·33-s − 0.171·34-s + 0.493·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{6} \cdot 41^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{6} \cdot 41^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 7 | | \( 1 \) |
| 41 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 - T + p T^{2} - T^{3} + p^{2} T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 3 | $S_4\times C_2$ | \( 1 - T + T^{2} - p T^{3} + p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) |
| 13 | $S_4\times C_2$ | \( 1 + 9 T + 61 T^{2} + 249 T^{3} + 61 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + T + 29 T^{2} + 61 T^{3} + 29 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 13 T + 109 T^{2} + 555 T^{3} + 109 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 11 T + 101 T^{2} + 525 T^{3} + 101 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 10 T + 103 T^{2} - 572 T^{3} + 103 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 2 T + 77 T^{2} - 100 T^{3} + 77 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 3 T - 11 T^{2} + 277 T^{3} - 11 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 9 T + 25 T^{2} + 61 T^{3} + 25 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 7 T + 101 T^{2} - 445 T^{3} + 101 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 2 T + 119 T^{2} - 284 T^{3} + 119 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{3} \) |
| 61 | $S_4\times C_2$ | \( 1 - 4 T + 147 T^{2} - 336 T^{3} + 147 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 10 T + 193 T^{2} - 1140 T^{3} + 193 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 14 T + 197 T^{2} + 1964 T^{3} + 197 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 4 T + 135 T^{2} - 192 T^{3} + 135 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 14 T + 269 T^{2} - 2156 T^{3} + 269 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 2 T + 181 T^{2} - 388 T^{3} + 181 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 5 T + T^{2} - 911 T^{3} + p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 27 T + 493 T^{2} + 5699 T^{3} + 493 p T^{4} + 27 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.412029314110026175949421277819, −8.202405041730449973946826595242, −8.013895832810679007695376676576, −7.83867745020134996769628499545, −7.50746772565269887436584758510, −7.43126675549857140350527137595, −7.00054484989763590400389636460, −6.66745847174153822443668159451, −6.54211987417189412491932952262, −6.10654978789367376166885911470, −5.70284724374799198204905315437, −5.49232980663547576763895340246, −5.10779803960385738453534432341, −4.77813060699874174911354764206, −4.49645323795288868304564155992, −4.34833196089490003545012580519, −4.00427073390273818881907577066, −3.94926631395448961168327333737, −3.88765320249711006521781359838, −2.98325163329545008342103371485, −2.85123229080014921608369648573, −2.36230765223698651342811300968, −2.28814750638184948713660897127, −2.18472405961480177973500692731, −0.882468315462357779645774155532, 0, 0, 0,
0.882468315462357779645774155532, 2.18472405961480177973500692731, 2.28814750638184948713660897127, 2.36230765223698651342811300968, 2.85123229080014921608369648573, 2.98325163329545008342103371485, 3.88765320249711006521781359838, 3.94926631395448961168327333737, 4.00427073390273818881907577066, 4.34833196089490003545012580519, 4.49645323795288868304564155992, 4.77813060699874174911354764206, 5.10779803960385738453534432341, 5.49232980663547576763895340246, 5.70284724374799198204905315437, 6.10654978789367376166885911470, 6.54211987417189412491932952262, 6.66745847174153822443668159451, 7.00054484989763590400389636460, 7.43126675549857140350527137595, 7.50746772565269887436584758510, 7.83867745020134996769628499545, 8.013895832810679007695376676576, 8.202405041730449973946826595242, 8.412029314110026175949421277819
