L(s) = 1 | + 3-s + 2·5-s + 7-s + 9-s − 11-s + 2·13-s + 2·15-s + 6·17-s + 8·19-s + 21-s − 4·23-s − 25-s + 27-s + 2·29-s − 8·31-s − 33-s + 2·35-s + 6·37-s + 2·39-s + 6·41-s − 8·43-s + 2·45-s − 4·47-s + 49-s + 6·51-s + 10·53-s − 2·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.894·5-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.554·13-s + 0.516·15-s + 1.45·17-s + 1.83·19-s + 0.218·21-s − 0.834·23-s − 1/5·25-s + 0.192·27-s + 0.371·29-s − 1.43·31-s − 0.174·33-s + 0.338·35-s + 0.986·37-s + 0.320·39-s + 0.937·41-s − 1.21·43-s + 0.298·45-s − 0.583·47-s + 1/7·49-s + 0.840·51-s + 1.37·53-s − 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.289489553\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.289489553\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.520545243905015036617036411555, −7.69218427092622331467806163431, −7.36798314755470064921824933611, −6.05653516599518644681913317851, −5.65137065952180457375635800225, −4.82812768894369153607831538493, −3.68185644395261774273324479187, −3.02266388667526657105580156467, −1.94650440796263178211892860611, −1.13567723679308448242547585751,
1.13567723679308448242547585751, 1.94650440796263178211892860611, 3.02266388667526657105580156467, 3.68185644395261774273324479187, 4.82812768894369153607831538493, 5.65137065952180457375635800225, 6.05653516599518644681913317851, 7.36798314755470064921824933611, 7.69218427092622331467806163431, 8.520545243905015036617036411555