| L(s) = 1 | − 2i·7-s − 3·9-s − 5.29i·11-s + 2i·13-s + 5.29i·19-s − 6i·23-s + (1 + 5.29i)29-s − 5.29i·31-s − 10.5·43-s − 10.5·47-s + 3·49-s + 2i·53-s + 10.5i·61-s + 6i·63-s + 10i·67-s + ⋯ |
| L(s) = 1 | − 0.755i·7-s − 9-s − 1.59i·11-s + 0.554i·13-s + 1.21i·19-s − 1.25i·23-s + (0.185 + 0.982i)29-s − 0.950i·31-s − 1.61·43-s − 1.54·47-s + 0.428·49-s + 0.274i·53-s + 1.35i·61-s + 0.755i·63-s + 1.22i·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.961 - 0.273i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.961 - 0.273i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2055272496\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2055272496\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 29 | \( 1 + (-1 - 5.29i)T \) |
| good | 3 | \( 1 + 3T^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 5.29iT - 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 5.29iT - 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 31 | \( 1 + 5.29iT - 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 + 10.5T + 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 10.5iT - 61T^{2} \) |
| 67 | \( 1 - 10iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 - 5.29iT - 79T^{2} \) |
| 83 | \( 1 + 2iT - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + 10.5T + 97T^{2} \) |
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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
−8.539814525314577797112340331736, −7.72477934555918858659334270161, −6.72473956160134338454146786962, −6.07455479071190180493695527544, −5.40066971698413015609284899917, −4.30915572308539315346766286731, −3.49820118871128597957381202540, −2.73165813297041686320480685229, −1.32580939648423843555364605560, −0.06421872792595876817780059628,
1.73180742308726177443244139453, 2.64257527391498410904923987564, 3.44143151704029813485736118225, 4.78593197816589960468296934106, 5.17986085176006247368094666872, 6.13053619662874370377258252697, 6.90492772780020432886819229047, 7.72437624985030391934944423878, 8.439266612387759537273848991842, 9.203680114007339817675403989684
