| L(s) = 1 | − 8·41-s − 2·81-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + 251-s + ⋯ |
| L(s) = 1 | − 8·41-s − 2·81-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + 251-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4351404503\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4351404503\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 41 | $C_1$ | \( ( 1 + T )^{8} \) |
| 43 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
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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
−6.41473516732947081626218693252, −6.26241272749764046438391669309, −5.81497657749248190330483104458, −5.63017924837728775204799129432, −5.46083967693233716720656857596, −5.45490015785996835775925748502, −5.03045363450171316178310636004, −5.02880261823992864665981694581, −4.69652154913408887139214636706, −4.41164773945431332497170434520, −4.40442741252701725036055526623, −4.24170447864360398614841771417, −3.55069198096291171232947161275, −3.54382545189060467677852852507, −3.37693534274857280233526153234, −3.28993053408557069037493646195, −3.15620969394984185239991623342, −2.67783678238172333166591511944, −2.34654076833839008292226682875, −2.02394635151992678945856159291, −1.99642760280359098530146809054, −1.62408912239523120760558423085, −1.25049665581412941974710719706, −1.19846593088660010545087102603, −0.24654595920807105591038488511,
0.24654595920807105591038488511, 1.19846593088660010545087102603, 1.25049665581412941974710719706, 1.62408912239523120760558423085, 1.99642760280359098530146809054, 2.02394635151992678945856159291, 2.34654076833839008292226682875, 2.67783678238172333166591511944, 3.15620969394984185239991623342, 3.28993053408557069037493646195, 3.37693534274857280233526153234, 3.54382545189060467677852852507, 3.55069198096291171232947161275, 4.24170447864360398614841771417, 4.40442741252701725036055526623, 4.41164773945431332497170434520, 4.69652154913408887139214636706, 5.02880261823992864665981694581, 5.03045363450171316178310636004, 5.45490015785996835775925748502, 5.46083967693233716720656857596, 5.63017924837728775204799129432, 5.81497657749248190330483104458, 6.26241272749764046438391669309, 6.41473516732947081626218693252
