L(s) = 1 | + 2-s − 2·3-s − 2·6-s − 8-s + 3·9-s − 2·11-s − 16-s − 4·17-s + 3·18-s − 2·19-s − 2·22-s + 2·24-s − 4·27-s + 4·33-s − 4·34-s − 2·38-s + 41-s − 43-s + 2·48-s − 49-s + 8·51-s − 4·54-s + 4·57-s + 59-s + 64-s + 4·66-s − 67-s + ⋯ |
L(s) = 1 | + 2-s − 2·3-s − 2·6-s − 8-s + 3·9-s − 2·11-s − 16-s − 4·17-s + 3·18-s − 2·19-s − 2·22-s + 2·24-s − 4·27-s + 4·33-s − 4·34-s − 2·38-s + 41-s − 43-s + 2·48-s − 49-s + 8·51-s − 4·54-s + 4·57-s + 59-s + 64-s + 4·66-s − 67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.07231841195\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.07231841195\) |
\(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | | \( 1 \) |
good | 7 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_1$ | \( ( 1 + T )^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 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_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 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_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
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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
−9.910023944342792120838150622733, −9.221672433493689494542748373485, −8.931788012563920906567232087794, −8.484988356238490620612424190935, −8.092650685060006443180412257156, −7.52731592307891002553681013779, −6.91336963670413770661341706546, −6.62827361643327361832839757443, −6.42028049841365022151502090050, −6.10830087856580903077164341736, −5.36970526932147481326199900939, −5.31283448836759103443258226680, −4.60099810867796520080950289180, −4.58211161765208124931401182762, −4.18826557465494761172551114441, −3.72573923948335484107591889285, −2.53060944909435527588667559895, −2.44656023537320876925877225840, −1.77606184994408508858255127027, −0.18068456797024635616045494218,
0.18068456797024635616045494218, 1.77606184994408508858255127027, 2.44656023537320876925877225840, 2.53060944909435527588667559895, 3.72573923948335484107591889285, 4.18826557465494761172551114441, 4.58211161765208124931401182762, 4.60099810867796520080950289180, 5.31283448836759103443258226680, 5.36970526932147481326199900939, 6.10830087856580903077164341736, 6.42028049841365022151502090050, 6.62827361643327361832839757443, 6.91336963670413770661341706546, 7.52731592307891002553681013779, 8.092650685060006443180412257156, 8.484988356238490620612424190935, 8.931788012563920906567232087794, 9.221672433493689494542748373485, 9.910023944342792120838150622733