| L(s) = 1 | − 3.62i·5-s + 0.712i·7-s − 3.62·11-s − 3.02·13-s − 5.59i·17-s − i·19-s − 2.37·23-s − 8.14·25-s − 1.44i·29-s + 1.02i·31-s + 2.58·35-s − 3.02·37-s − 2.01i·41-s + 7.78i·43-s + 6.97·47-s + ⋯ |
| L(s) = 1 | − 1.62i·5-s + 0.269i·7-s − 1.09·11-s − 0.838·13-s − 1.35i·17-s − 0.229i·19-s − 0.494·23-s − 1.62·25-s − 0.269i·29-s + 0.184i·31-s + 0.436·35-s − 0.497·37-s − 0.314i·41-s + 1.18i·43-s + 1.01·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.169 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.08685427456\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08685427456\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + iT \) |
| good | 5 | \( 1 + 3.62iT - 5T^{2} \) |
| 7 | \( 1 - 0.712iT - 7T^{2} \) |
| 11 | \( 1 + 3.62T + 11T^{2} \) |
| 13 | \( 1 + 3.02T + 13T^{2} \) |
| 17 | \( 1 + 5.59iT - 17T^{2} \) |
| 23 | \( 1 + 2.37T + 23T^{2} \) |
| 29 | \( 1 + 1.44iT - 29T^{2} \) |
| 31 | \( 1 - 1.02iT - 31T^{2} \) |
| 37 | \( 1 + 3.02T + 37T^{2} \) |
| 41 | \( 1 + 2.01iT - 41T^{2} \) |
| 43 | \( 1 - 7.78iT - 43T^{2} \) |
| 47 | \( 1 - 6.97T + 47T^{2} \) |
| 53 | \( 1 - 2.40iT - 53T^{2} \) |
| 59 | \( 1 + 5.80T + 59T^{2} \) |
| 61 | \( 1 + 11.8T + 61T^{2} \) |
| 67 | \( 1 - 5.42iT - 67T^{2} \) |
| 71 | \( 1 - 1.87T + 71T^{2} \) |
| 73 | \( 1 - 5.49T + 73T^{2} \) |
| 79 | \( 1 - 7.07iT - 79T^{2} \) |
| 83 | \( 1 - 5.20T + 83T^{2} \) |
| 89 | \( 1 - 2.68iT - 89T^{2} \) |
| 97 | \( 1 - 12.9T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.326372072417034483334788860042, −7.75190122829482906730409411035, −7.14014952368657855254345743524, −5.98226964617801216472640710346, −5.25590215184087906913909720157, −4.88508335137163046681290612251, −4.18477773769324286893186424665, −2.90294160556771866042294637348, −2.17535569860638036514779382802, −0.951135848284301155699712840254,
0.02500793903243852481174444570, 1.88624925235895752471397679174, 2.58507883511856333167961513313, 3.38375179834297189290211750453, 4.09883121827182565005203635548, 5.14804922505075758107934591085, 5.95798756051925329119137533384, 6.54135599002053159092689561352, 7.41550835591703588353574300508, 7.66716471325674914118027479768
