| L(s) = 1 | + 5-s − 2·7-s − 9-s − 2·11-s + 17-s − 2·23-s + 25-s − 2·35-s + 43-s − 45-s + 47-s + 49-s − 2·55-s + 61-s + 2·63-s + 73-s + 4·77-s + 4·83-s + 85-s + 2·99-s − 2·101-s − 2·115-s − 2·119-s + 121-s + 2·125-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 5-s − 2·7-s − 9-s − 2·11-s + 17-s − 2·23-s + 25-s − 2·35-s + 43-s − 45-s + 47-s + 49-s − 2·55-s + 61-s + 2·63-s + 73-s + 4·77-s + 4·83-s + 85-s + 2·99-s − 2·101-s − 2·115-s − 2·119-s + 121-s + 2·125-s + 127-s + 131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2085136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2085136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6377214842\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6377214842\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 19 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_1$ | \( ( 1 - T )^{4} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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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
−10.07357511585902444549712042577, −9.450894102198111153581618043208, −9.378425387187765761140308637221, −8.829971732467341798668091817329, −8.159199594605645473307960865990, −8.013120238370716009915483029140, −7.64468821262505167670420497366, −6.94543881930385196867850818075, −6.61236037191190504224677213673, −6.08622087369974846490588521189, −5.86864876691273257795639999140, −5.45530975981446613030679895909, −5.23553128078275267156677108279, −4.47181732868480573484273247992, −3.76164804416741179238826528102, −3.28912761803733197402161722223, −2.91333630617745562481150830755, −2.40290751585530755382738396254, −2.04427352570763004950562669902, −0.61502814383840666277453031149,
0.61502814383840666277453031149, 2.04427352570763004950562669902, 2.40290751585530755382738396254, 2.91333630617745562481150830755, 3.28912761803733197402161722223, 3.76164804416741179238826528102, 4.47181732868480573484273247992, 5.23553128078275267156677108279, 5.45530975981446613030679895909, 5.86864876691273257795639999140, 6.08622087369974846490588521189, 6.61236037191190504224677213673, 6.94543881930385196867850818075, 7.64468821262505167670420497366, 8.013120238370716009915483029140, 8.159199594605645473307960865990, 8.829971732467341798668091817329, 9.378425387187765761140308637221, 9.450894102198111153581618043208, 10.07357511585902444549712042577
