| L(s) = 1 | + 1.21i·5-s + (0.355 + 2.62i)7-s + 4.06·11-s − 1.42·13-s − 4.74·17-s − 4.84·19-s + 6.03i·23-s + 3.51·25-s − 3.45·29-s − 3.15i·31-s + (−3.19 + 0.432i)35-s + 9.24i·37-s − 1.24·41-s − 3.23i·43-s − 9.43·47-s + ⋯ |
| L(s) = 1 | + 0.544i·5-s + (0.134 + 0.990i)7-s + 1.22·11-s − 0.396·13-s − 1.15·17-s − 1.11·19-s + 1.25i·23-s + 0.703·25-s − 0.641·29-s − 0.566i·31-s + (−0.539 + 0.0731i)35-s + 1.51i·37-s − 0.194·41-s − 0.492i·43-s − 1.37·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 - 0.217i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.976 - 0.217i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9280936385\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9280936385\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.355 - 2.62i)T \) |
| good | 5 | \( 1 - 1.21iT - 5T^{2} \) |
| 11 | \( 1 - 4.06T + 11T^{2} \) |
| 13 | \( 1 + 1.42T + 13T^{2} \) |
| 17 | \( 1 + 4.74T + 17T^{2} \) |
| 19 | \( 1 + 4.84T + 19T^{2} \) |
| 23 | \( 1 - 6.03iT - 23T^{2} \) |
| 29 | \( 1 + 3.45T + 29T^{2} \) |
| 31 | \( 1 + 3.15iT - 31T^{2} \) |
| 37 | \( 1 - 9.24iT - 37T^{2} \) |
| 41 | \( 1 + 1.24T + 41T^{2} \) |
| 43 | \( 1 + 3.23iT - 43T^{2} \) |
| 47 | \( 1 + 9.43T + 47T^{2} \) |
| 53 | \( 1 - 2.44T + 53T^{2} \) |
| 59 | \( 1 + 10.2iT - 59T^{2} \) |
| 61 | \( 1 - 10.1T + 61T^{2} \) |
| 67 | \( 1 - 6.51iT - 67T^{2} \) |
| 71 | \( 1 + 2.14iT - 71T^{2} \) |
| 73 | \( 1 - 12.6iT - 73T^{2} \) |
| 79 | \( 1 + 10.6T + 79T^{2} \) |
| 83 | \( 1 + 16.3iT - 83T^{2} \) |
| 89 | \( 1 + 14.9T + 89T^{2} \) |
| 97 | \( 1 - 11.4iT - 97T^{2} \) |
| 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
−8.713089273540949780865768164924, −8.261487238869507407800413792776, −7.09142785616793165644949369751, −6.63678732932849792746055268438, −5.95168446690618028221695443933, −5.04941478748665620840765309115, −4.22510909817483897967675774002, −3.33564304423093821302403836018, −2.38415819296558488981807764310, −1.59722282234956096392065330718,
0.25727600761256058070694697146, 1.39025269247369201248608914933, 2.38144814900612838287053076082, 3.68212260055143709304549749897, 4.36886873100233469834094693092, 4.80612797897637173526037024710, 6.01612258903847361277606181046, 6.79132783123079688502545608699, 7.14574495256564272290832424630, 8.313287981167952273529439383510
