L(s) = 1 | + 2·3-s − 2·5-s − 7-s + 9-s + 2·13-s − 4·15-s + 6·17-s + 6·19-s − 2·21-s + 4·23-s − 25-s − 4·27-s − 8·31-s + 2·35-s + 8·37-s + 4·39-s + 6·41-s + 8·43-s − 2·45-s + 49-s + 12·51-s − 4·53-s + 12·57-s + 6·59-s − 2·61-s − 63-s − 4·65-s + ⋯ |
L(s) = 1 | + 1.15·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s + 0.554·13-s − 1.03·15-s + 1.45·17-s + 1.37·19-s − 0.436·21-s + 0.834·23-s − 1/5·25-s − 0.769·27-s − 1.43·31-s + 0.338·35-s + 1.31·37-s + 0.640·39-s + 0.937·41-s + 1.21·43-s − 0.298·45-s + 1/7·49-s + 1.68·51-s − 0.549·53-s + 1.58·57-s + 0.781·59-s − 0.256·61-s − 0.125·63-s − 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.231216288\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.231216288\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−9.372760439871449542670551342185, −8.406405509142369471222003907281, −7.64959686513607590170159758025, −7.40739365135633793675611792599, −6.05419653120887467329096156931, −5.19831326896907265809056215806, −3.81662601696213052961590423089, −3.47427752772640011549825658590, −2.55414931813316300438399962237, −1.00745331396629620659974078069,
1.00745331396629620659974078069, 2.55414931813316300438399962237, 3.47427752772640011549825658590, 3.81662601696213052961590423089, 5.19831326896907265809056215806, 6.05419653120887467329096156931, 7.40739365135633793675611792599, 7.64959686513607590170159758025, 8.406405509142369471222003907281, 9.372760439871449542670551342185