| L(s) = 1 | + 2·2-s − 3-s + 3·4-s − 2·6-s + 4·8-s + 11-s − 3·12-s + 5·16-s + 17-s − 19-s + 2·22-s − 4·24-s − 25-s + 27-s + 6·32-s − 33-s + 2·34-s − 2·38-s − 2·41-s − 2·43-s + 3·44-s − 5·48-s − 49-s − 2·50-s − 51-s + 2·54-s + 57-s + ⋯ |
| L(s) = 1 | + 2·2-s − 3-s + 3·4-s − 2·6-s + 4·8-s + 11-s − 3·12-s + 5·16-s + 17-s − 19-s + 2·22-s − 4·24-s − 25-s + 27-s + 6·32-s − 33-s + 2·34-s − 2·38-s − 2·41-s − 2·43-s + 3·44-s − 5·48-s − 49-s − 2·50-s − 51-s + 2·54-s + 57-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
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.104206366\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.104206366\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_1$ | \( ( 1 - T )^{4} \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.959161577990900682258233736298, −9.933395296642072710366356307055, −9.448101999875661682021079428351, −8.469573491655852700780784530475, −8.200496083450288031370306366030, −7.984628726720632745776299382721, −7.16388383461423760372204156393, −6.81681449248100472005855428732, −6.44396145670729245783316950852, −6.38262839318434213840635797776, −5.61333046101326014519586695911, −5.49763570087490325524422019751, −4.89004857543489799639344870133, −4.71444878540678821472544809858, −3.96685342899043908544354573930, −3.61783465249980083822586076150, −3.26824092855577981618522693483, −2.54001284021414971578418490426, −1.81663886755901956018562637406, −1.35917532374469826631182757309,
1.35917532374469826631182757309, 1.81663886755901956018562637406, 2.54001284021414971578418490426, 3.26824092855577981618522693483, 3.61783465249980083822586076150, 3.96685342899043908544354573930, 4.71444878540678821472544809858, 4.89004857543489799639344870133, 5.49763570087490325524422019751, 5.61333046101326014519586695911, 6.38262839318434213840635797776, 6.44396145670729245783316950852, 6.81681449248100472005855428732, 7.16388383461423760372204156393, 7.984628726720632745776299382721, 8.200496083450288031370306366030, 8.469573491655852700780784530475, 9.448101999875661682021079428351, 9.933395296642072710366356307055, 9.959161577990900682258233736298
