| L(s) = 1 | − 2·3-s + 2·5-s + 9-s − 4·15-s + 16-s + 25-s + 2·27-s + 2·45-s − 2·48-s − 2·75-s + 2·80-s − 4·81-s + 4·109-s − 2·121-s − 2·125-s + 127-s + 131-s + 4·135-s + 137-s + 139-s + 144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-s + ⋯ |
| L(s) = 1 | − 2·3-s + 2·5-s + 9-s − 4·15-s + 16-s + 25-s + 2·27-s + 2·45-s − 2·48-s − 2·75-s + 2·80-s − 4·81-s + 4·109-s − 2·121-s − 2·125-s + 127-s + 131-s + 4·135-s + 137-s + 139-s + 144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4964584647\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4964584647\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 19 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 23 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 31 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 53 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 61 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
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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
−7.71546850894808407275465067059, −7.19367491245394962018639464830, −7.12523073467722465423152838236, −7.06892040177764118705906923877, −6.41694160162284171093223295223, −6.34927474447112696564101843963, −6.27063349225231279058608907105, −6.05526125403350144727956635115, −5.81323282971528958859886389574, −5.54555463523792179981987746177, −5.51504386370411204034742997640, −5.26355169963314242333493287683, −4.76418900317356564866494229436, −4.74432837941686459881135014539, −4.70840255113185615258939622988, −4.02943667866464755173379376161, −3.65925848659690912855683389582, −3.58242554572332453256505996505, −3.10681241598429729348555652662, −2.63487817523132710027597905007, −2.54939778975243280944115154810, −2.15150059135229144708006282839, −1.63469389138249823907523463923, −1.37810325116112342368004411780, −0.857268764279335774323015240523,
0.857268764279335774323015240523, 1.37810325116112342368004411780, 1.63469389138249823907523463923, 2.15150059135229144708006282839, 2.54939778975243280944115154810, 2.63487817523132710027597905007, 3.10681241598429729348555652662, 3.58242554572332453256505996505, 3.65925848659690912855683389582, 4.02943667866464755173379376161, 4.70840255113185615258939622988, 4.74432837941686459881135014539, 4.76418900317356564866494229436, 5.26355169963314242333493287683, 5.51504386370411204034742997640, 5.54555463523792179981987746177, 5.81323282971528958859886389574, 6.05526125403350144727956635115, 6.27063349225231279058608907105, 6.34927474447112696564101843963, 6.41694160162284171093223295223, 7.06892040177764118705906923877, 7.12523073467722465423152838236, 7.19367491245394962018639464830, 7.71546850894808407275465067059
