| L(s) = 1 | − 2·3-s − 3·5-s + 7-s − 3·9-s − 2·11-s − 2·13-s + 6·15-s + 2·17-s − 6·19-s − 2·21-s + 5·25-s + 14·27-s + 9·29-s − 31-s + 4·33-s − 3·35-s − 10·37-s + 4·39-s − 3·41-s − 3·43-s + 9·45-s − 11·47-s − 6·49-s − 4·51-s − 3·53-s + 6·55-s + 12·57-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.34·5-s + 0.377·7-s − 9-s − 0.603·11-s − 0.554·13-s + 1.54·15-s + 0.485·17-s − 1.37·19-s − 0.436·21-s + 25-s + 2.69·27-s + 1.67·29-s − 0.179·31-s + 0.696·33-s − 0.507·35-s − 1.64·37-s + 0.640·39-s − 0.468·41-s − 0.457·43-s + 1.34·45-s − 1.60·47-s − 6/7·49-s − 0.560·51-s − 0.412·53-s + 0.809·55-s + 1.58·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529984 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + T - 30 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 3 T - 34 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 11 T + 74 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - T - 70 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 9 T + 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 10 T + 11 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 13 T + 72 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
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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
−10.43314165639347719582902835866, −10.00542970929772758021743164874, −9.212346144177994265292567663126, −8.699245434152593349296847277192, −8.362682670475705107528823623482, −8.182539007613774978172420930244, −7.54297387751119883840500787058, −7.20874132291086813603661315875, −6.36867670511174030911188506353, −6.34680427516949457631538116206, −5.74122411813187522605003405952, −5.01161062926751993932665747937, −4.71576316896358109364951279152, −4.60077667820527999883254340837, −3.40608782404698359156344392919, −3.21903736313216218490155977672, −2.50341705440163373610399575474, −1.45242600029716425330608173542, 0, 0,
1.45242600029716425330608173542, 2.50341705440163373610399575474, 3.21903736313216218490155977672, 3.40608782404698359156344392919, 4.60077667820527999883254340837, 4.71576316896358109364951279152, 5.01161062926751993932665747937, 5.74122411813187522605003405952, 6.34680427516949457631538116206, 6.36867670511174030911188506353, 7.20874132291086813603661315875, 7.54297387751119883840500787058, 8.182539007613774978172420930244, 8.362682670475705107528823623482, 8.699245434152593349296847277192, 9.212346144177994265292567663126, 10.00542970929772758021743164874, 10.43314165639347719582902835866
