| L(s) = 1 | + 2-s + 4-s − 5-s − 7-s + 8-s − 10-s + 6·11-s + 2·13-s − 14-s + 16-s − 4·19-s − 20-s + 6·22-s + 9·23-s + 25-s + 2·26-s − 28-s + 3·29-s − 4·31-s + 32-s + 35-s + 8·37-s − 4·38-s − 40-s − 3·41-s + 8·43-s + 6·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s + 0.353·8-s − 0.316·10-s + 1.80·11-s + 0.554·13-s − 0.267·14-s + 1/4·16-s − 0.917·19-s − 0.223·20-s + 1.27·22-s + 1.87·23-s + 1/5·25-s + 0.392·26-s − 0.188·28-s + 0.557·29-s − 0.718·31-s + 0.176·32-s + 0.169·35-s + 1.31·37-s − 0.648·38-s − 0.158·40-s − 0.468·41-s + 1.21·43-s + 0.904·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.394075810\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.394075810\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 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
−10.46895616919797963639308649861, −9.228530484906492722876848200952, −8.717378153471938511526709867807, −7.42407483081609479054268881777, −6.62750192113583511421649895960, −5.99605261483170121408204274618, −4.63365408594636780088580922329, −3.90981816402000462440548769015, −2.95626008175642733558542036272, −1.29915321561212133045480179839,
1.29915321561212133045480179839, 2.95626008175642733558542036272, 3.90981816402000462440548769015, 4.63365408594636780088580922329, 5.99605261483170121408204274618, 6.62750192113583511421649895960, 7.42407483081609479054268881777, 8.717378153471938511526709867807, 9.228530484906492722876848200952, 10.46895616919797963639308649861
