L(s) = 1 | + 2·2-s + 3-s + 3·4-s + 2·6-s + 4·8-s + 3·12-s + 5·16-s − 3·17-s − 2·19-s + 4·24-s + 25-s − 27-s + 6·32-s − 6·34-s − 4·38-s − 41-s + 2·43-s + 5·48-s − 49-s + 2·50-s − 3·51-s − 2·54-s − 2·57-s + 59-s + 7·64-s − 9·68-s + 2·73-s + ⋯ |
L(s) = 1 | + 2·2-s + 3-s + 3·4-s + 2·6-s + 4·8-s + 3·12-s + 5·16-s − 3·17-s − 2·19-s + 4·24-s + 25-s − 27-s + 6·32-s − 6·34-s − 4·38-s − 41-s + 2·43-s + 5·48-s − 49-s + 2·50-s − 3·51-s − 2·54-s − 2·57-s + 59-s + 7·64-s − 9·68-s + 2·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1871424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1871424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(5.049609999\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.049609999\) |
\(L(1)\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.05738317743343404393516529408, −9.645283325981158141300416177522, −9.035382161108632739797915950459, −8.603574794593597963786900524688, −8.375799794996604349316021392341, −8.022572272044741461334092658294, −7.31059235504777561464214283150, −6.89272122826084991012093978262, −6.56649810600448946036840716606, −6.49605851361768181737660137189, −5.61742885590298475158585292665, −5.51862397367057283459929811540, −4.54941426375695327588876820909, −4.47637269167217473955573615766, −4.13935682905790601537629219185, −3.60316751573377103670795345528, −2.90370896257695148726069880142, −2.52308499792742952572811717848, −2.19004075749737114168407515878, −1.67247916555918633457811860114,
1.67247916555918633457811860114, 2.19004075749737114168407515878, 2.52308499792742952572811717848, 2.90370896257695148726069880142, 3.60316751573377103670795345528, 4.13935682905790601537629219185, 4.47637269167217473955573615766, 4.54941426375695327588876820909, 5.51862397367057283459929811540, 5.61742885590298475158585292665, 6.49605851361768181737660137189, 6.56649810600448946036840716606, 6.89272122826084991012093978262, 7.31059235504777561464214283150, 8.022572272044741461334092658294, 8.375799794996604349316021392341, 8.603574794593597963786900524688, 9.035382161108632739797915950459, 9.645283325981158141300416177522, 10.05738317743343404393516529408