| L(s) = 1 | + i·3-s + 4.60i·7-s − 9-s + 2.60·11-s + 2.60i·13-s + 2i·17-s + 19-s − 4.60·21-s + 2i·23-s − i·27-s + 4.60·29-s + 4·31-s + 2.60i·33-s + 10.6i·37-s − 2.60·39-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 1.74i·7-s − 0.333·9-s + 0.785·11-s + 0.722i·13-s + 0.485i·17-s + 0.229·19-s − 1.00·21-s + 0.417i·23-s − 0.192i·27-s + 0.855·29-s + 0.718·31-s + 0.453i·33-s + 1.74i·37-s − 0.417·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.899894688\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.899894688\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 - 4.60iT - 7T^{2} \) |
| 11 | \( 1 - 2.60T + 11T^{2} \) |
| 13 | \( 1 - 2.60iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 23 | \( 1 - 2iT - 23T^{2} \) |
| 29 | \( 1 - 4.60T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 10.6iT - 37T^{2} \) |
| 41 | \( 1 + 0.605T + 41T^{2} \) |
| 43 | \( 1 + 3.39iT - 43T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 9.21T + 59T^{2} \) |
| 61 | \( 1 - 7.21T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 - 5.21T + 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 3.21iT - 83T^{2} \) |
| 89 | \( 1 - 0.605T + 89T^{2} \) |
| 97 | \( 1 - 9.39iT - 97T^{2} \) |
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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
−8.596882241669409243764873223127, −7.980308206304230218696710145926, −6.73585975390349637861066712683, −6.29319903035531770072425059463, −5.51747304993038244274421415654, −4.85556304382750168198636763453, −4.05993220308650087301257348602, −3.13523256540552728695406380586, −2.38450692860476508683403465132, −1.38850619339090398990940024761,
0.56213506729579740972018908432, 1.13081903582611908980782006597, 2.37537333992468358445477490984, 3.41046269792789815684681433124, 4.05719202516120323601925636133, 4.84890887175608268155220443936, 5.76620051252535272443789361066, 6.70971472239696729018720155381, 7.00877786063695622531713951640, 7.74834021150912345075683713364
