| L(s) = 1 | − 0.176·2-s + 3-s − 1.96·4-s − 4.00·5-s − 0.176·6-s − 7-s + 0.701·8-s + 9-s + 0.708·10-s − 1.64·11-s − 1.96·12-s − 2.11·13-s + 0.176·14-s − 4.00·15-s + 3.81·16-s − 0.176·18-s − 1.44·19-s + 7.88·20-s − 21-s + 0.291·22-s + 6.00·23-s + 0.701·24-s + 11.0·25-s + 0.374·26-s + 27-s + 1.96·28-s + 0.508·29-s + ⋯ |
| L(s) = 1 | − 0.124·2-s + 0.577·3-s − 0.984·4-s − 1.79·5-s − 0.0721·6-s − 0.377·7-s + 0.248·8-s + 0.333·9-s + 0.223·10-s − 0.497·11-s − 0.568·12-s − 0.587·13-s + 0.0472·14-s − 1.03·15-s + 0.953·16-s − 0.0416·18-s − 0.330·19-s + 1.76·20-s − 0.218·21-s + 0.0621·22-s + 1.25·23-s + 0.143·24-s + 2.20·25-s + 0.0733·26-s + 0.192·27-s + 0.372·28-s + 0.0943·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6069 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6069 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 0.176T + 2T^{2} \) |
| 5 | \( 1 + 4.00T + 5T^{2} \) |
| 11 | \( 1 + 1.64T + 11T^{2} \) |
| 13 | \( 1 + 2.11T + 13T^{2} \) |
| 19 | \( 1 + 1.44T + 19T^{2} \) |
| 23 | \( 1 - 6.00T + 23T^{2} \) |
| 29 | \( 1 - 0.508T + 29T^{2} \) |
| 31 | \( 1 - 4.13T + 31T^{2} \) |
| 37 | \( 1 + 1.06T + 37T^{2} \) |
| 41 | \( 1 - 9.40T + 41T^{2} \) |
| 43 | \( 1 + 8.11T + 43T^{2} \) |
| 47 | \( 1 + 9.98T + 47T^{2} \) |
| 53 | \( 1 - 10.9T + 53T^{2} \) |
| 59 | \( 1 + 7.35T + 59T^{2} \) |
| 61 | \( 1 - 11.2T + 61T^{2} \) |
| 67 | \( 1 + 6.33T + 67T^{2} \) |
| 71 | \( 1 - 4.59T + 71T^{2} \) |
| 73 | \( 1 - 10.0T + 73T^{2} \) |
| 79 | \( 1 + 2.14T + 79T^{2} \) |
| 83 | \( 1 - 7.32T + 83T^{2} \) |
| 89 | \( 1 - 5.50T + 89T^{2} \) |
| 97 | \( 1 - 14.4T + 97T^{2} \) |
| 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
−7.915763943918763212138713582015, −7.25987403785015485913643145192, −6.56790580985550735012831004055, −5.20345032786640423959448551209, −4.70984272744255671358479967056, −3.97610195131336213791616737375, −3.37432803623806821659269954330, −2.62133808613398283556574168536, −0.953269429926442961922493914270, 0,
0.953269429926442961922493914270, 2.62133808613398283556574168536, 3.37432803623806821659269954330, 3.97610195131336213791616737375, 4.70984272744255671358479967056, 5.20345032786640423959448551209, 6.56790580985550735012831004055, 7.25987403785015485913643145192, 7.915763943918763212138713582015
