| L(s) = 1 | + 2·2-s − 3-s + 4-s − 3·5-s − 2·6-s − 2·7-s − 2·8-s + 3·9-s − 6·10-s − 3·11-s − 12-s − 4·13-s − 4·14-s + 3·15-s − 4·16-s − 6·17-s + 6·18-s + 20·19-s − 3·20-s + 2·21-s − 6·22-s + 9·23-s + 2·24-s + 4·25-s − 8·26-s − 8·27-s − 2·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 1/2·4-s − 1.34·5-s − 0.816·6-s − 0.755·7-s − 0.707·8-s + 9-s − 1.89·10-s − 0.904·11-s − 0.288·12-s − 1.10·13-s − 1.06·14-s + 0.774·15-s − 16-s − 1.45·17-s + 1.41·18-s + 4.58·19-s − 0.670·20-s + 0.436·21-s − 1.27·22-s + 1.87·23-s + 0.408·24-s + 4/5·25-s − 1.56·26-s − 1.53·27-s − 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.095306656\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.095306656\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| good | 5 | $C_2$$\times$$C_2^2$ | \( ( 1 + 3 T + p T^{2} )^{2}( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} ) \) |
| 11 | $D_4\times C_2$ | \( 1 + 3 T - 7 T^{2} - 18 T^{3} + 36 T^{4} - 18 p T^{5} - 7 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 + 3 T + 28 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{4} \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 - 9 T + p T^{2} )^{2}( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} ) \) |
| 29 | $D_4\times C_2$ | \( 1 - 6 T + 2 T^{2} + 144 T^{3} - 729 T^{4} + 144 p T^{5} + 2 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 41 | $D_4\times C_2$ | \( 1 + 15 T + 95 T^{2} + 720 T^{3} + 5994 T^{4} + 720 p T^{5} + 95 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + T - 11 T^{2} - 74 T^{3} - 1748 T^{4} - 74 p T^{5} - 11 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 + 6 T + 82 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 3 T - 37 T^{2} + 216 T^{3} - 1896 T^{4} + 216 p T^{5} - 37 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 11 T + 43 T^{2} + 484 T^{3} - 5018 T^{4} + 484 p T^{5} + 43 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + 13 T + p T^{2} )^{2}( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} ) \) |
| 71 | $D_{4}$ | \( ( 1 - 3 T + 70 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 - 7 T + 84 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 7 T - 47 T^{2} - 434 T^{3} - 896 T^{4} - 434 p T^{5} - 47 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 12 T + 74 T^{2} - 1152 T^{3} - 13941 T^{4} - 1152 p T^{5} + 74 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 18 T + 226 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + T - 119 T^{2} - 74 T^{3} + 4894 T^{4} - 74 p T^{5} - 119 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
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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
−9.943271288692848233802268113406, −9.854577104533250915313846011950, −9.185892151627262300803817088652, −9.087993171069103384190256023281, −8.985285160020422183853974825269, −8.418965819437021315515620696375, −7.77686893599899371508110623579, −7.69647613661721236000456706668, −7.67447534995630322848161254471, −7.07303048009666643407173618507, −7.04015083891063419778197537518, −6.67740331843246986413126198964, −6.36220790983528955629297556857, −5.74801752248677133077945527113, −5.50179419082868554959466102205, −5.15364717351957514406832532988, −4.92274678794252482475757130607, −4.70354047588690341765218080075, −4.46452392562003441426142369778, −3.79691459021131243112805053186, −3.30560809605902297363921093595, −3.24916221363599159764289039433, −2.98073045426211311221302203718, −2.10732594047403510097966828064, −0.815943745770869659414733200548,
0.815943745770869659414733200548, 2.10732594047403510097966828064, 2.98073045426211311221302203718, 3.24916221363599159764289039433, 3.30560809605902297363921093595, 3.79691459021131243112805053186, 4.46452392562003441426142369778, 4.70354047588690341765218080075, 4.92274678794252482475757130607, 5.15364717351957514406832532988, 5.50179419082868554959466102205, 5.74801752248677133077945527113, 6.36220790983528955629297556857, 6.67740331843246986413126198964, 7.04015083891063419778197537518, 7.07303048009666643407173618507, 7.67447534995630322848161254471, 7.69647613661721236000456706668, 7.77686893599899371508110623579, 8.418965819437021315515620696375, 8.985285160020422183853974825269, 9.087993171069103384190256023281, 9.185892151627262300803817088652, 9.854577104533250915313846011950, 9.943271288692848233802268113406
