| L(s) = 1 | − 2-s + 4-s + 1.23·5-s + 2.85·7-s − 8-s − 1.23·10-s − 6.23·13-s − 2.85·14-s + 16-s + 6.23·17-s + 5·19-s + 1.23·20-s − 7.61·23-s − 3.47·25-s + 6.23·26-s + 2.85·28-s − 7.09·29-s − 7·31-s − 32-s − 6.23·34-s + 3.52·35-s + 9·37-s − 5·38-s − 1.23·40-s − 3.38·41-s − 0.708·43-s + 7.61·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.552·5-s + 1.07·7-s − 0.353·8-s − 0.390·10-s − 1.72·13-s − 0.762·14-s + 0.250·16-s + 1.51·17-s + 1.14·19-s + 0.276·20-s − 1.58·23-s − 0.694·25-s + 1.22·26-s + 0.539·28-s − 1.31·29-s − 1.25·31-s − 0.176·32-s − 1.06·34-s + 0.596·35-s + 1.47·37-s − 0.811·38-s − 0.195·40-s − 0.528·41-s − 0.108·43-s + 1.12·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6534 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6534 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 1.23T + 5T^{2} \) |
| 7 | \( 1 - 2.85T + 7T^{2} \) |
| 13 | \( 1 + 6.23T + 13T^{2} \) |
| 17 | \( 1 - 6.23T + 17T^{2} \) |
| 19 | \( 1 - 5T + 19T^{2} \) |
| 23 | \( 1 + 7.61T + 23T^{2} \) |
| 29 | \( 1 + 7.09T + 29T^{2} \) |
| 31 | \( 1 + 7T + 31T^{2} \) |
| 37 | \( 1 - 9T + 37T^{2} \) |
| 41 | \( 1 + 3.38T + 41T^{2} \) |
| 43 | \( 1 + 0.708T + 43T^{2} \) |
| 47 | \( 1 + 1.76T + 47T^{2} \) |
| 53 | \( 1 + 10.0T + 53T^{2} \) |
| 59 | \( 1 + 9.32T + 59T^{2} \) |
| 61 | \( 1 - 3.47T + 61T^{2} \) |
| 67 | \( 1 + 6.09T + 67T^{2} \) |
| 71 | \( 1 + 4.52T + 71T^{2} \) |
| 73 | \( 1 + 4.23T + 73T^{2} \) |
| 79 | \( 1 - 15T + 79T^{2} \) |
| 83 | \( 1 - 3.76T + 83T^{2} \) |
| 89 | \( 1 + 4.23T + 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
| 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
−7.72370923130816785459042619590, −7.38408925130446220413581572823, −6.19799887698531923805394199067, −5.49358067916238941310937559670, −5.04988042357323360343730998885, −3.97220004667936209604688150228, −2.96498537498752631367970420253, −2.00038788647118960174637348752, −1.46245474308659192288051088787, 0,
1.46245474308659192288051088787, 2.00038788647118960174637348752, 2.96498537498752631367970420253, 3.97220004667936209604688150228, 5.04988042357323360343730998885, 5.49358067916238941310937559670, 6.19799887698531923805394199067, 7.38408925130446220413581572823, 7.72370923130816785459042619590
