| L(s) = 1 | + 2-s + 4-s + 2·5-s + 2.56·7-s + 8-s + 2·10-s + 2.56·11-s − 5.68·13-s + 2.56·14-s + 16-s − 0.561·17-s + 0.561·19-s + 2·20-s + 2.56·22-s − 3.68·23-s − 25-s − 5.68·26-s + 2.56·28-s + 7.12·29-s − 0.876·31-s + 32-s − 0.561·34-s + 5.12·35-s − 37-s + 0.561·38-s + 2·40-s + 2.87·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.894·5-s + 0.968·7-s + 0.353·8-s + 0.632·10-s + 0.772·11-s − 1.57·13-s + 0.684·14-s + 0.250·16-s − 0.136·17-s + 0.128·19-s + 0.447·20-s + 0.546·22-s − 0.768·23-s − 0.200·25-s − 1.11·26-s + 0.484·28-s + 1.32·29-s − 0.157·31-s + 0.176·32-s − 0.0963·34-s + 0.865·35-s − 0.164·37-s + 0.0910·38-s + 0.316·40-s + 0.449·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.822174801\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.822174801\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 5 | \( 1 - 2T + 5T^{2} \) |
| 7 | \( 1 - 2.56T + 7T^{2} \) |
| 11 | \( 1 - 2.56T + 11T^{2} \) |
| 13 | \( 1 + 5.68T + 13T^{2} \) |
| 17 | \( 1 + 0.561T + 17T^{2} \) |
| 19 | \( 1 - 0.561T + 19T^{2} \) |
| 23 | \( 1 + 3.68T + 23T^{2} \) |
| 29 | \( 1 - 7.12T + 29T^{2} \) |
| 31 | \( 1 + 0.876T + 31T^{2} \) |
| 41 | \( 1 - 2.87T + 41T^{2} \) |
| 43 | \( 1 + 7.12T + 43T^{2} \) |
| 47 | \( 1 - 6.24T + 47T^{2} \) |
| 53 | \( 1 - 2.56T + 53T^{2} \) |
| 59 | \( 1 - 9.12T + 59T^{2} \) |
| 61 | \( 1 - 4.24T + 61T^{2} \) |
| 67 | \( 1 + 14.2T + 67T^{2} \) |
| 71 | \( 1 + 14.2T + 71T^{2} \) |
| 73 | \( 1 + 0.561T + 73T^{2} \) |
| 79 | \( 1 + 6T + 79T^{2} \) |
| 83 | \( 1 + 13.9T + 83T^{2} \) |
| 89 | \( 1 - 15.9T + 89T^{2} \) |
| 97 | \( 1 - 11.1T + 97T^{2} \) |
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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
−10.43582316363845337161231782326, −9.857800476957259042721346919601, −8.817740248924404818828940750889, −7.74084034610088728247134103433, −6.85408294300160265772614092842, −5.86961259376518556524014837057, −5.00726630021782130617327699168, −4.20239100090993711110256832913, −2.64507034270072575979041934969, −1.66356127585176219027364750538,
1.66356127585176219027364750538, 2.64507034270072575979041934969, 4.20239100090993711110256832913, 5.00726630021782130617327699168, 5.86961259376518556524014837057, 6.85408294300160265772614092842, 7.74084034610088728247134103433, 8.817740248924404818828940750889, 9.857800476957259042721346919601, 10.43582316363845337161231782326
