| L(s) = 1 | − 1.59i·3-s − 5-s + 1.92·7-s + 0.463·9-s + (−3.09 − 1.18i)11-s − 0.937i·13-s + 1.59i·15-s + 1.34i·17-s − 4.67·19-s − 3.07i·21-s − 1.65i·23-s + 25-s − 5.51i·27-s − 4.09i·29-s − 6.07i·31-s + ⋯ |
| L(s) = 1 | − 0.919i·3-s − 0.447·5-s + 0.728·7-s + 0.154·9-s + (−0.934 − 0.356i)11-s − 0.260i·13-s + 0.411i·15-s + 0.325i·17-s − 1.07·19-s − 0.670i·21-s − 0.345i·23-s + 0.200·25-s − 1.06i·27-s − 0.761i·29-s − 1.09i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.912 + 0.408i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.912 + 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.055566443\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.055566443\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + (3.09 + 1.18i)T \) |
| good | 3 | \( 1 + 1.59iT - 3T^{2} \) |
| 7 | \( 1 - 1.92T + 7T^{2} \) |
| 13 | \( 1 + 0.937iT - 13T^{2} \) |
| 17 | \( 1 - 1.34iT - 17T^{2} \) |
| 19 | \( 1 + 4.67T + 19T^{2} \) |
| 23 | \( 1 + 1.65iT - 23T^{2} \) |
| 29 | \( 1 + 4.09iT - 29T^{2} \) |
| 31 | \( 1 + 6.07iT - 31T^{2} \) |
| 37 | \( 1 - 1.36T + 37T^{2} \) |
| 41 | \( 1 - 0.518iT - 41T^{2} \) |
| 43 | \( 1 - 0.102T + 43T^{2} \) |
| 47 | \( 1 + 7.10iT - 47T^{2} \) |
| 53 | \( 1 + 9.66T + 53T^{2} \) |
| 59 | \( 1 - 1.63iT - 59T^{2} \) |
| 61 | \( 1 - 4.26iT - 61T^{2} \) |
| 67 | \( 1 + 2.72iT - 67T^{2} \) |
| 71 | \( 1 + 11.9iT - 71T^{2} \) |
| 73 | \( 1 - 9.77iT - 73T^{2} \) |
| 79 | \( 1 - 0.844T + 79T^{2} \) |
| 83 | \( 1 + 7.44T + 83T^{2} \) |
| 89 | \( 1 - 4.91T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
| 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
−8.595670789625675487070245745257, −8.013077307687959155659490736791, −7.60287489906112803475310538037, −6.61270741817365805710281000657, −5.87064416404920856289833930092, −4.80415092719207388651604412335, −4.00370189834391752202037172953, −2.65813597429696669550396609950, −1.75720582638286758164419719471, −0.38725325510252956077952632674,
1.61401899875613452466177674549, 2.91649355149391014349013937746, 3.96396119204044086689924455711, 4.76409342543255912065842519331, 5.18356794906810191895712935297, 6.49816456119804848025890259625, 7.40687012675200176283350879383, 8.093020202605674217497453238461, 8.910879004115237195182444720062, 9.656776375569177941869714730778
