| 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 − 41-s + 43-s + 3·44-s + 5·48-s − 2·50-s − 51-s − 2·54-s − 57-s + 2·59-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 − 41-s + 43-s + 3·44-s + 5·48-s − 2·50-s − 51-s − 2·54-s − 57-s + 2·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(8.359830485\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.359830485\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 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
−8.868320329685531293846956959772, −8.425826204894481936475446853936, −8.167578558847364501277142845723, −7.70676547718515562990107387411, −7.26686065516604024798251323788, −7.05462246201832031123260113220, −6.56741018628835899063694737414, −6.26367210459782307283985806503, −5.82980397746158007887450123493, −5.72036448764215642598741643293, −4.85488169532320179437857847262, −4.77685260662310705366140561125, −4.21831153950315463935850209968, −3.77282126933970821665267478568, −3.70429763108106388955683202623, −3.15085597277977048280272391480, −2.55812812276892009212820654145, −2.22635946106678392607468345056, −1.91556826351457052678913916276, −1.25223699249503463699553132193,
1.25223699249503463699553132193, 1.91556826351457052678913916276, 2.22635946106678392607468345056, 2.55812812276892009212820654145, 3.15085597277977048280272391480, 3.70429763108106388955683202623, 3.77282126933970821665267478568, 4.21831153950315463935850209968, 4.77685260662310705366140561125, 4.85488169532320179437857847262, 5.72036448764215642598741643293, 5.82980397746158007887450123493, 6.26367210459782307283985806503, 6.56741018628835899063694737414, 7.05462246201832031123260113220, 7.26686065516604024798251323788, 7.70676547718515562990107387411, 8.167578558847364501277142845723, 8.425826204894481936475446853936, 8.868320329685531293846956959772
