| L(s) = 1 | − 4·2-s + 8·4-s + 12·7-s − 8·8-s − 12·11-s − 24·13-s − 48·14-s − 4·16-s + 12·17-s + 48·22-s − 24·23-s + 96·26-s + 96·28-s − 56·31-s + 32·32-s − 48·34-s − 24·37-s + 120·41-s − 144·43-s − 96·44-s + 96·46-s − 72·47-s + 72·49-s − 192·52-s − 48·53-s − 96·56-s + 188·61-s + ⋯ |
| L(s) = 1 | − 2·2-s + 2·4-s + 12/7·7-s − 8-s − 1.09·11-s − 1.84·13-s − 3.42·14-s − 1/4·16-s + 0.705·17-s + 2.18·22-s − 1.04·23-s + 3.69·26-s + 24/7·28-s − 1.80·31-s + 32-s − 1.41·34-s − 0.648·37-s + 2.92·41-s − 3.34·43-s − 2.18·44-s + 2.08·46-s − 1.53·47-s + 1.46·49-s − 3.69·52-s − 0.905·53-s − 1.71·56-s + 3.08·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1014061069\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1014061069\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 660 T^{3} + 6014 T^{4} - 660 p^{2} T^{5} + 72 p^{4} T^{6} - 12 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p^{2} T^{2} )^{4} \) |
| 13 | $D_4\times C_2$ | \( 1 + 24 T + 288 T^{2} + 5136 T^{3} + 89567 T^{4} + 5136 p^{2} T^{5} + 288 p^{4} T^{6} + 24 p^{6} T^{7} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 2388 T^{3} + 71102 T^{4} - 2388 p^{2} T^{5} + 72 p^{4} T^{6} - 12 p^{6} T^{7} + p^{8} T^{8} \) |
| 19 | $C_2^3$ | \( 1 - 884 T^{2} + 1110 p^{2} T^{4} - 884 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 24 T + 288 T^{2} + 8592 T^{3} + 227087 T^{4} + 8592 p^{2} T^{5} + 288 p^{4} T^{6} + 24 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 1780 T^{2} + 1708998 T^{4} - 1780 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 28 T + 1632 T^{2} + 28 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 24 T + 288 T^{2} + 34512 T^{3} + 4130927 T^{4} + 34512 p^{2} T^{5} + 288 p^{4} T^{6} + 24 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 60 T + 3776 T^{2} - 60 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 144 T + 10368 T^{2} + 611856 T^{3} + 30348002 T^{4} + 611856 p^{2} T^{5} + 10368 p^{4} T^{6} + 144 p^{6} T^{7} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 72 T + 2592 T^{2} + 48240 T^{3} - 1470721 T^{4} + 48240 p^{2} T^{5} + 2592 p^{4} T^{6} + 72 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 48 T + 1152 T^{2} + 127920 T^{3} + 14183714 T^{4} + 127920 p^{2} T^{5} + 1152 p^{4} T^{6} + 48 p^{6} T^{7} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 9154 T^{2} + 39927075 T^{4} - 9154 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 94 T + 9435 T^{2} - 94 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 204 T + 20808 T^{2} + 1426164 T^{3} + 91488158 T^{4} + 1426164 p^{2} T^{5} + 20808 p^{4} T^{6} + 204 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 30 T + 4907 T^{2} + 30 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 240 T + 28800 T^{2} + 3004080 T^{3} + 261683234 T^{4} + 3004080 p^{2} T^{5} + 28800 p^{4} T^{6} + 240 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 11888 T^{2} + 113126754 T^{4} - 11888 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 192 T + 18432 T^{2} - 1875648 T^{3} + 182572322 T^{4} - 1875648 p^{2} T^{5} + 18432 p^{4} T^{6} - 192 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 5584 T^{2} + 23163810 T^{4} - 5584 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 336 T + 56448 T^{2} + 6934368 T^{3} + 725763647 T^{4} + 6934368 p^{2} T^{5} + 56448 p^{4} T^{6} + 336 p^{6} T^{7} + p^{8} T^{8} \) |
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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
−6.79531281830025940422683903792, −6.69381481903123957756240737864, −6.41434999632264782329819406405, −5.92073088845652032769297673348, −5.72366398763436279799222648203, −5.49446016635323147898372835644, −5.45683682793145306853338859540, −5.11529150354433478043571596243, −4.87188397908826669470577385503, −4.69306277397416474814800200835, −4.57918490596121375719445643866, −4.23901322192465169780087229569, −3.75668482730444091612643058919, −3.69215176320571219013372596696, −3.48416911977597981515360989811, −2.74341890662993674102046281576, −2.62037517415430741726653077083, −2.57197469373663628950071689541, −2.32909958729042507874298490569, −1.67478721468211453707793783585, −1.50383179527476599512305968263, −1.47135235014055696421039470546, −1.18826770428959051576050973582, −0.21325082772583066551736121508, −0.17989961003331118318113266760,
0.17989961003331118318113266760, 0.21325082772583066551736121508, 1.18826770428959051576050973582, 1.47135235014055696421039470546, 1.50383179527476599512305968263, 1.67478721468211453707793783585, 2.32909958729042507874298490569, 2.57197469373663628950071689541, 2.62037517415430741726653077083, 2.74341890662993674102046281576, 3.48416911977597981515360989811, 3.69215176320571219013372596696, 3.75668482730444091612643058919, 4.23901322192465169780087229569, 4.57918490596121375719445643866, 4.69306277397416474814800200835, 4.87188397908826669470577385503, 5.11529150354433478043571596243, 5.45683682793145306853338859540, 5.49446016635323147898372835644, 5.72366398763436279799222648203, 5.92073088845652032769297673348, 6.41434999632264782329819406405, 6.69381481903123957756240737864, 6.79531281830025940422683903792
