| L(s) = 1 | + 4-s − 4·9-s − 2·11-s + 2·19-s − 2·31-s − 4·36-s − 2·44-s − 4·49-s + 2·76-s − 4·79-s + 10·81-s + 2·89-s + 8·99-s − 2·101-s + 121-s − 2·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s − 8·171-s + ⋯ |
| L(s) = 1 | + 4-s − 4·9-s − 2·11-s + 2·19-s − 2·31-s − 4·36-s − 2·44-s − 4·49-s + 2·76-s − 4·79-s + 10·81-s + 2·89-s + 8·99-s − 2·101-s + 121-s − 2·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s − 8·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 79^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 79^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.002633790010\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.002633790010\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 79 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 2 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 3 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 11 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 13 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 19 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 23 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 31 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 67 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 89 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 97 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
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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
−6.73784434404710056966631771293, −6.26670983509683467230463290888, −6.21390436680707159320100339821, −6.20097218535694920580075245881, −5.83607964520597216631779331238, −5.65148346643809418653664292336, −5.51977287740885538396895729465, −5.18687773721832427217101701105, −5.15969279836730106289144859601, −5.00512660658087552940120361320, −4.87389026170591000067396495481, −4.41600050465062195330488982680, −4.08831533484336784572638961913, −3.47830670147833745669285728142, −3.46518184360324496114732800617, −3.42441867362929581086425294668, −3.13526401956997833404193791357, −2.84189087946226129861056641964, −2.53064405916649520857153098566, −2.47728253103133410443785772920, −2.41038003220776168509411392583, −1.86647075045885345822920846539, −1.47711377691477051913384929197, −1.15282602885632469257823728687, −0.02426107055162404012020213940,
0.02426107055162404012020213940, 1.15282602885632469257823728687, 1.47711377691477051913384929197, 1.86647075045885345822920846539, 2.41038003220776168509411392583, 2.47728253103133410443785772920, 2.53064405916649520857153098566, 2.84189087946226129861056641964, 3.13526401956997833404193791357, 3.42441867362929581086425294668, 3.46518184360324496114732800617, 3.47830670147833745669285728142, 4.08831533484336784572638961913, 4.41600050465062195330488982680, 4.87389026170591000067396495481, 5.00512660658087552940120361320, 5.15969279836730106289144859601, 5.18687773721832427217101701105, 5.51977287740885538396895729465, 5.65148346643809418653664292336, 5.83607964520597216631779331238, 6.20097218535694920580075245881, 6.21390436680707159320100339821, 6.26670983509683467230463290888, 6.73784434404710056966631771293
