| L(s) = 1 | + i·3-s + (−0.726 − 2.11i)5-s − 4.05i·7-s − 9-s − 0.985·11-s − 4.94i·13-s + (2.11 − 0.726i)15-s − 4.52i·17-s + 2.60·19-s + 4.05·21-s − 3.53i·23-s + (−3.94 + 3.07i)25-s − i·27-s + 7.59·29-s − 3.28·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (−0.324 − 0.945i)5-s − 1.53i·7-s − 0.333·9-s − 0.297·11-s − 1.37i·13-s + (0.546 − 0.187i)15-s − 1.09i·17-s + 0.597·19-s + 0.885·21-s − 0.737i·23-s + (−0.789 + 0.614i)25-s − 0.192i·27-s + 1.41·29-s − 0.589·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.945 + 0.324i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.945 + 0.324i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.168182621\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.168182621\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.726 + 2.11i)T \) |
| good | 7 | \( 1 + 4.05iT - 7T^{2} \) |
| 11 | \( 1 + 0.985T + 11T^{2} \) |
| 13 | \( 1 + 4.94iT - 13T^{2} \) |
| 17 | \( 1 + 4.52iT - 17T^{2} \) |
| 19 | \( 1 - 2.60T + 19T^{2} \) |
| 23 | \( 1 + 3.53iT - 23T^{2} \) |
| 29 | \( 1 - 7.59T + 29T^{2} \) |
| 31 | \( 1 + 3.28T + 31T^{2} \) |
| 37 | \( 1 + 0.945iT - 37T^{2} \) |
| 41 | \( 1 + 0.568T + 41T^{2} \) |
| 43 | \( 1 - 8.45iT - 43T^{2} \) |
| 47 | \( 1 + 2.60iT - 47T^{2} \) |
| 53 | \( 1 - 0.229iT - 53T^{2} \) |
| 59 | \( 1 - 9.10T + 59T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 - 8.45iT - 67T^{2} \) |
| 71 | \( 1 + 1.43T + 71T^{2} \) |
| 73 | \( 1 + 11.9iT - 73T^{2} \) |
| 79 | \( 1 - 3.28T + 79T^{2} \) |
| 83 | \( 1 + 9.89iT - 83T^{2} \) |
| 89 | \( 1 + 12.3T + 89T^{2} \) |
| 97 | \( 1 - 3.23iT - 97T^{2} \) |
| 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
−8.037169934120814147002491221201, −7.64363133901054510511999913657, −6.81566472931555814947586042212, −5.71152482048527083741713659856, −4.91580937259486285417909122559, −4.48868947184065417177414116252, −3.56778738258108060876252119273, −2.81373919142952999708236449747, −1.09073304727314019575066454253, −0.37998610173395063241020068851,
1.65732293935353401716738906275, 2.39200131736231924995304642253, 3.18773624230507423504849854436, 4.14069246971246725356100490381, 5.29056103557544347593363796035, 5.97332435199928257645628690459, 6.62238195357486009286982030701, 7.25138318920282039742611555035, 8.134703801754617421436765517514, 8.679859433526011658569171882112
