| 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
| \(L(1)\) |
\(\approx\) |
\(0.6685034769\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6685034769\) |
| \(L(\frac{3}{2})\) |
|
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
−8.338906641868561516316533859195, −7.923011725876952771073731127613, −7.81698743367539954901972329680, −7.77494686559317854283764393927, −7.37692648358527437884318697690, −6.67043765492159671967928505318, −6.49264706546436716596580184498, −6.46206055683122965932303134810, −5.95612042529863387173062281709, −5.90988370336443668837317880771, −5.26307996422272041614928370399, −5.16625062277484521195012928593, −5.03729536728359239158677515489, −4.43093079687699160127310534176, −4.40279779028563590865029681150, −4.19915599037581549161596894791, −3.68433833529564192528452710165, −3.17905533826100351543722360593, −3.00123712922808238790919492155, −2.91509849934404604267717942191, −2.28566201773003561363661374490, −1.71957703631108711214885126995, −0.879967975859371124733281420881, −0.73963652845832953910809386062, −0.46127665438387725214585069244,
0.46127665438387725214585069244, 0.73963652845832953910809386062, 0.879967975859371124733281420881, 1.71957703631108711214885126995, 2.28566201773003561363661374490, 2.91509849934404604267717942191, 3.00123712922808238790919492155, 3.17905533826100351543722360593, 3.68433833529564192528452710165, 4.19915599037581549161596894791, 4.40279779028563590865029681150, 4.43093079687699160127310534176, 5.03729536728359239158677515489, 5.16625062277484521195012928593, 5.26307996422272041614928370399, 5.90988370336443668837317880771, 5.95612042529863387173062281709, 6.46206055683122965932303134810, 6.49264706546436716596580184498, 6.67043765492159671967928505318, 7.37692648358527437884318697690, 7.77494686559317854283764393927, 7.81698743367539954901972329680, 7.923011725876952771073731127613, 8.338906641868561516316533859195
