| L(s) = 1 | − 4·2-s + 8·4-s − 12·5-s − 8·8-s − 3·9-s + 48·10-s − 18·13-s − 4·16-s − 18·17-s + 12·18-s − 96·20-s + 74·25-s + 72·26-s − 10·29-s + 32·32-s + 72·34-s − 24·36-s − 6·37-s + 96·40-s − 6·41-s + 36·45-s + 13·49-s − 296·50-s − 144·52-s − 2·53-s + 40·58-s − 64·64-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 4·4-s − 5.36·5-s − 2.82·8-s − 9-s + 15.1·10-s − 4.99·13-s − 16-s − 4.36·17-s + 2.82·18-s − 21.4·20-s + 74/5·25-s + 14.1·26-s − 1.85·29-s + 5.65·32-s + 12.3·34-s − 4·36-s − 0.986·37-s + 15.1·40-s − 0.937·41-s + 5.36·45-s + 13/7·49-s − 41.8·50-s − 19.9·52-s − 0.274·53-s + 5.25·58-s − 8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| good | 5 | $C_2^2$ | \( ( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 + 18 T^{2} + 203 T^{4} + 18 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 9 T + 44 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^3$ | \( 1 - 35 T^{2} + 864 T^{4} - 35 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^3$ | \( 1 + 30 T^{2} + 371 T^{4} + 30 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 5 T - 4 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 35 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 3 T + 44 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 - 35 T^{2} - 624 T^{4} - 35 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + 91 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + T - 52 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 115 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 95 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 53 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 138 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 21 T + 220 T^{2} + 21 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 149 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 139 T^{2} + 12432 T^{4} - 139 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 15 T + 164 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 3 T + 100 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−9.201777011342048515482415272553, −9.017003666081641634336198264152, −8.695252479940453419913592429805, −8.392180430447137745710025233043, −8.356146977247064240702626260333, −7.928608073454293163366949066974, −7.86387797654245248854361540894, −7.56185125510803068444613010408, −7.48152746099534992270864424385, −7.03293301250601471266251954068, −6.95287434774973908929836873107, −6.92931409694225584808319047439, −6.92029228221117370305331795787, −6.04111805822899657810248528764, −5.21971357394230672962812450717, −4.96636417586955352086330618606, −4.74205698022061536421782953276, −4.58309559017506750217701112909, −4.00911864210596074217135043914, −4.00474979300844267647547283737, −3.99029659077505534818667743828, −2.81488133743212904072203226784, −2.71545191740239012378783046545, −2.50556179857420631260033446627, −1.84937841768072233237056378663, 0, 0, 0, 0,
1.84937841768072233237056378663, 2.50556179857420631260033446627, 2.71545191740239012378783046545, 2.81488133743212904072203226784, 3.99029659077505534818667743828, 4.00474979300844267647547283737, 4.00911864210596074217135043914, 4.58309559017506750217701112909, 4.74205698022061536421782953276, 4.96636417586955352086330618606, 5.21971357394230672962812450717, 6.04111805822899657810248528764, 6.92029228221117370305331795787, 6.92931409694225584808319047439, 6.95287434774973908929836873107, 7.03293301250601471266251954068, 7.48152746099534992270864424385, 7.56185125510803068444613010408, 7.86387797654245248854361540894, 7.928608073454293163366949066974, 8.356146977247064240702626260333, 8.392180430447137745710025233043, 8.695252479940453419913592429805, 9.017003666081641634336198264152, 9.201777011342048515482415272553
