| L(s) = 1 | + 2·2-s − 3-s + 3·4-s − 2·6-s + 4·8-s − 2·11-s − 3·12-s + 5·16-s + 17-s + 2·19-s − 4·22-s − 4·24-s − 25-s + 27-s + 6·32-s + 2·33-s + 2·34-s + 4·38-s + 41-s − 2·43-s − 6·44-s − 5·48-s − 49-s − 2·50-s − 51-s + 2·54-s − 2·57-s + ⋯ |
| L(s) = 1 | + 2·2-s − 3-s + 3·4-s − 2·6-s + 4·8-s − 2·11-s − 3·12-s + 5·16-s + 17-s + 2·19-s − 4·22-s − 4·24-s − 25-s + 27-s + 6·32-s + 2·33-s + 2·34-s + 4·38-s + 41-s − 2·43-s − 6·44-s − 5·48-s − 49-s − 2·50-s − 51-s + 2·54-s − 2·57-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
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.808107118\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.808107118\) |
| \(L(1)\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 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_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 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_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 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 | $C_1$ | \( ( 1 - T )^{4} \) |
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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
−10.23964752453507542887075538181, −9.864479568909705467475486134678, −9.488287464231363701618401999262, −8.504353236912691631248548878968, −8.119422975356664011586898569850, −7.70703983486328758816680507904, −7.35058419738252049834606447643, −7.16891998039505855706247864239, −6.42355668609849813766621903921, −6.01995005770348045884318893545, −5.60612457448574239441537681935, −5.50356808072668493089014499172, −4.90040998567246129818852550110, −4.85566921462868626831538754276, −4.13335733852875406146776655894, −3.32379899449556343526853567375, −3.17504849756498667230766547579, −2.70968048426587108429779118541, −1.96570688891125341170354393523, −1.19095235372130077440993126435,
1.19095235372130077440993126435, 1.96570688891125341170354393523, 2.70968048426587108429779118541, 3.17504849756498667230766547579, 3.32379899449556343526853567375, 4.13335733852875406146776655894, 4.85566921462868626831538754276, 4.90040998567246129818852550110, 5.50356808072668493089014499172, 5.60612457448574239441537681935, 6.01995005770348045884318893545, 6.42355668609849813766621903921, 7.16891998039505855706247864239, 7.35058419738252049834606447643, 7.70703983486328758816680507904, 8.119422975356664011586898569850, 8.504353236912691631248548878968, 9.488287464231363701618401999262, 9.864479568909705467475486134678, 10.23964752453507542887075538181
