| L(s) = 1 | − 2·2-s + 2·4-s − 8·7-s − 4·8-s − 3·9-s + 16·14-s + 8·16-s + 20·17-s + 6·18-s − 4·23-s − 6·25-s − 16·28-s + 8·31-s − 8·32-s − 40·34-s − 6·36-s + 10·41-s + 8·46-s + 12·47-s + 30·49-s + 12·50-s + 32·56-s − 16·62-s + 24·63-s + 8·64-s + 40·68-s − 24·71-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 3.02·7-s − 1.41·8-s − 9-s + 4.27·14-s + 2·16-s + 4.85·17-s + 1.41·18-s − 0.834·23-s − 6/5·25-s − 3.02·28-s + 1.43·31-s − 1.41·32-s − 6.85·34-s − 36-s + 1.56·41-s + 1.17·46-s + 1.75·47-s + 30/7·49-s + 1.69·50-s + 4.27·56-s − 2.03·62-s + 3.02·63-s + 64-s + 4.85·68-s − 2.84·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2242125845\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2242125845\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| good | 5 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \) |
| 11 | $C_2^3$ | \( 1 + 13 T^{2} + 48 T^{4} + 13 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - T^{2} + p^{2} T^{4} )( 1 + 23 T^{2} + p^{2} T^{4} ) \) |
| 17 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 37 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 2 T - 19 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 5 T - 16 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 - 35 T^{2} - 624 T^{4} - 35 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 59 | $C_2^3$ | \( 1 + 117 T^{2} + 10208 T^{4} + 117 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} )( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} ) \) |
| 67 | $C_2^3$ | \( 1 + 125 T^{2} + 11136 T^{4} + 125 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 14 T + 117 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 150 T^{2} + 15611 T^{4} + 150 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 + T - 96 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−10.72570299456177282736360006327, −10.34326000840739451213141599018, −10.13251765185218936814101371061, −9.804990840672986192092784488114, −9.640827493928599083202869088148, −9.557197029839213090619546303397, −9.443369303386711787844930720142, −8.864233791200871868373160028945, −8.370991586526766644620930897732, −8.289693896764442110070953991204, −7.72766623118291055585766666734, −7.69991990660725264961026418096, −7.35564051714940854583092676864, −6.70322670223246431719646270721, −6.30334462187447895786850271729, −6.24321063915552237522606768556, −5.62945459857429237262547075789, −5.59999673995470352727593848927, −5.44469430329803475229840398074, −4.08380052048807056291574053817, −3.70685003664035112911932368740, −3.32942124861589429372848992180, −2.87052640885778066793988540552, −2.75148535640928773077850167267, −0.884195755745734177353933001764,
0.884195755745734177353933001764, 2.75148535640928773077850167267, 2.87052640885778066793988540552, 3.32942124861589429372848992180, 3.70685003664035112911932368740, 4.08380052048807056291574053817, 5.44469430329803475229840398074, 5.59999673995470352727593848927, 5.62945459857429237262547075789, 6.24321063915552237522606768556, 6.30334462187447895786850271729, 6.70322670223246431719646270721, 7.35564051714940854583092676864, 7.69991990660725264961026418096, 7.72766623118291055585766666734, 8.289693896764442110070953991204, 8.370991586526766644620930897732, 8.864233791200871868373160028945, 9.443369303386711787844930720142, 9.557197029839213090619546303397, 9.640827493928599083202869088148, 9.804990840672986192092784488114, 10.13251765185218936814101371061, 10.34326000840739451213141599018, 10.72570299456177282736360006327
