| L(s) = 1 | + 2·3-s − 3·9-s − 2·11-s + 12·17-s + 16·19-s − 25-s − 14·27-s − 4·33-s − 20·43-s − 10·49-s + 24·51-s + 32·57-s + 6·59-s − 2·67-s − 8·73-s − 2·75-s − 4·81-s + 12·83-s − 18·89-s − 14·97-s + 6·99-s + 12·107-s − 30·113-s + 3·121-s + 127-s − 40·129-s + 131-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 9-s − 0.603·11-s + 2.91·17-s + 3.67·19-s − 1/5·25-s − 2.69·27-s − 0.696·33-s − 3.04·43-s − 1.42·49-s + 3.36·51-s + 4.23·57-s + 0.781·59-s − 0.244·67-s − 0.936·73-s − 0.230·75-s − 4/9·81-s + 1.31·83-s − 1.90·89-s − 1.42·97-s + 0.603·99-s + 1.16·107-s − 2.82·113-s + 3/11·121-s + 0.0887·127-s − 3.52·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30976 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.737676634\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.737676634\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
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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
−10.11903216163362853339678805280, −9.813753565180213813561671136388, −9.610210113213273792621314474885, −8.901169398745525002958448579942, −8.132665807481574488057539990847, −7.948144712624414981153354525702, −7.60852684015633966980596452192, −6.87940646463052539617018989539, −5.72427671804102205390312745209, −5.44949075059971672760817509463, −5.11227132195889762323685903220, −3.47350326380949607246521494930, −3.27381985666804460223319703873, −2.88597193878705003332421314931, −1.40865503128588890763537566425,
1.40865503128588890763537566425, 2.88597193878705003332421314931, 3.27381985666804460223319703873, 3.47350326380949607246521494930, 5.11227132195889762323685903220, 5.44949075059971672760817509463, 5.72427671804102205390312745209, 6.87940646463052539617018989539, 7.60852684015633966980596452192, 7.948144712624414981153354525702, 8.132665807481574488057539990847, 8.901169398745525002958448579942, 9.610210113213273792621314474885, 9.813753565180213813561671136388, 10.11903216163362853339678805280
