L(s) = 1 | + 4-s − 2·5-s − 2·17-s − 2·20-s + 2·25-s + 4·29-s + 2·37-s − 4·41-s − 2·61-s − 64-s − 2·68-s + 2·73-s + 81-s + 4·85-s + 4·97-s + 2·100-s + 2·109-s − 4·113-s + 4·116-s + 127-s + 131-s + 137-s + 139-s − 8·145-s + 2·148-s + 149-s + 151-s + ⋯ |
L(s) = 1 | + 4-s − 2·5-s − 2·17-s − 2·20-s + 2·25-s + 4·29-s + 2·37-s − 4·41-s − 2·61-s − 64-s − 2·68-s + 2·73-s + 81-s + 4·85-s + 4·97-s + 2·100-s + 2·109-s − 4·113-s + 4·116-s + 127-s + 131-s + 137-s + 139-s − 8·145-s + 2·148-s + 149-s + 151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9978833281\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9978833281\) |
\(L(1)\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 7 | | \( 1 \) |
| 17 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
good | 3 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 5 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 11 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
| 31 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 61 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 73 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 79 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
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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
−6.32654930285682835395030353108, −6.17715472083869137951955940554, −6.15226271948719341569155687289, −5.90580615144908169331305987135, −5.26002006287524167896518932741, −5.20041136760701921358032342895, −4.94478206096547218130774118951, −4.85075914889955199496866068541, −4.76158875561994378183592410107, −4.45683931327154314116181802282, −4.22355377291641730510013127029, −4.03666501744667103644446276145, −3.94578985778106671440609392825, −3.61484624605464500043420853571, −3.16736292453920796230399277180, −3.10644338290444907650720087152, −3.07668220427799491294578385916, −2.67711051019348771809339368661, −2.55430590685711043479117431061, −2.04235347799218895192951728472, −1.97040737002892401057212561163, −1.76116675812638902489328841727, −1.17071985576115596186024613357, −0.893142184475783003192371305848, −0.45204804704840273107825280082,
0.45204804704840273107825280082, 0.893142184475783003192371305848, 1.17071985576115596186024613357, 1.76116675812638902489328841727, 1.97040737002892401057212561163, 2.04235347799218895192951728472, 2.55430590685711043479117431061, 2.67711051019348771809339368661, 3.07668220427799491294578385916, 3.10644338290444907650720087152, 3.16736292453920796230399277180, 3.61484624605464500043420853571, 3.94578985778106671440609392825, 4.03666501744667103644446276145, 4.22355377291641730510013127029, 4.45683931327154314116181802282, 4.76158875561994378183592410107, 4.85075914889955199496866068541, 4.94478206096547218130774118951, 5.20041136760701921358032342895, 5.26002006287524167896518932741, 5.90580615144908169331305987135, 6.15226271948719341569155687289, 6.17715472083869137951955940554, 6.32654930285682835395030353108