| L(s) = 1 | + 4·49-s − 81-s + 8·97-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 | + 4·49-s − 81-s + 8·97-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.779017518\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.779017518\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 + T^{4} \) |
| good | 5 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 97 | $C_1$ | \( ( 1 - T )^{8} \) |
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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
−6.35987649485421005494787342589, −5.99574931940502672455653576533, −5.93490095436040219595301790049, −5.86135806230783521844654220983, −5.79916754763116127285146233624, −5.12663337476553957180311472218, −5.10259930224557558556830228620, −5.02716164288572057464889281164, −4.82184414023189294510672141893, −4.57924482306293120338808032144, −4.24438376670352310000488130924, −4.00689566264957394810232429056, −3.90294273121176848437951031786, −3.63967306527139709749155052014, −3.54815237099680412111304605985, −3.15747861379236883038058783961, −2.91612516622658256786460100480, −2.65376650948268390370063819365, −2.50229553473516403073351853980, −2.14011945446793410127392067925, −1.86676712152264104142063758650, −1.83951326644066734521503688133, −1.21653681652877569593334382239, −0.848209316173399195777955746979, −0.68821966455674584709200266864,
0.68821966455674584709200266864, 0.848209316173399195777955746979, 1.21653681652877569593334382239, 1.83951326644066734521503688133, 1.86676712152264104142063758650, 2.14011945446793410127392067925, 2.50229553473516403073351853980, 2.65376650948268390370063819365, 2.91612516622658256786460100480, 3.15747861379236883038058783961, 3.54815237099680412111304605985, 3.63967306527139709749155052014, 3.90294273121176848437951031786, 4.00689566264957394810232429056, 4.24438376670352310000488130924, 4.57924482306293120338808032144, 4.82184414023189294510672141893, 5.02716164288572057464889281164, 5.10259930224557558556830228620, 5.12663337476553957180311472218, 5.79916754763116127285146233624, 5.86135806230783521844654220983, 5.93490095436040219595301790049, 5.99574931940502672455653576533, 6.35987649485421005494787342589
