| L(s) = 1 | − 1.15·2-s + 3-s − 0.662·4-s − 2.13·5-s − 1.15·6-s + 1.76·7-s + 3.07·8-s + 9-s + 2.46·10-s + 4.54·11-s − 0.662·12-s + 3.20·13-s − 2.04·14-s − 2.13·15-s − 2.23·16-s + 17-s − 1.15·18-s − 1.93·19-s + 1.41·20-s + 1.76·21-s − 5.25·22-s + 0.915·23-s + 3.07·24-s − 0.459·25-s − 3.70·26-s + 27-s − 1.17·28-s + ⋯ |
| L(s) = 1 | − 0.817·2-s + 0.577·3-s − 0.331·4-s − 0.952·5-s − 0.472·6-s + 0.668·7-s + 1.08·8-s + 0.333·9-s + 0.779·10-s + 1.36·11-s − 0.191·12-s + 0.888·13-s − 0.546·14-s − 0.550·15-s − 0.558·16-s + 0.242·17-s − 0.272·18-s − 0.444·19-s + 0.315·20-s + 0.385·21-s − 1.11·22-s + 0.190·23-s + 0.628·24-s − 0.0919·25-s − 0.726·26-s + 0.192·27-s − 0.221·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 157 | \( 1 + T \) |
| good | 2 | \( 1 + 1.15T + 2T^{2} \) |
| 5 | \( 1 + 2.13T + 5T^{2} \) |
| 7 | \( 1 - 1.76T + 7T^{2} \) |
| 11 | \( 1 - 4.54T + 11T^{2} \) |
| 13 | \( 1 - 3.20T + 13T^{2} \) |
| 19 | \( 1 + 1.93T + 19T^{2} \) |
| 23 | \( 1 - 0.915T + 23T^{2} \) |
| 29 | \( 1 + 5.24T + 29T^{2} \) |
| 31 | \( 1 + 7.36T + 31T^{2} \) |
| 37 | \( 1 - 0.773T + 37T^{2} \) |
| 41 | \( 1 + 0.531T + 41T^{2} \) |
| 43 | \( 1 + 4.03T + 43T^{2} \) |
| 47 | \( 1 + 4.48T + 47T^{2} \) |
| 53 | \( 1 + 1.87T + 53T^{2} \) |
| 59 | \( 1 - 1.10T + 59T^{2} \) |
| 61 | \( 1 + 9.49T + 61T^{2} \) |
| 67 | \( 1 - 6.65T + 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 + 4.36T + 73T^{2} \) |
| 79 | \( 1 - 7.69T + 79T^{2} \) |
| 83 | \( 1 + 7.68T + 83T^{2} \) |
| 89 | \( 1 + 17.2T + 89T^{2} \) |
| 97 | \( 1 - 5.11T + 97T^{2} \) |
| 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
−7.66817485336258764470169066222, −7.17413275748935259068827942192, −6.30880350363976726250300947195, −5.29583062804514499694552500206, −4.36290887622285205528535139990, −3.93378307595772972619775208564, −3.32423421847563609552663271984, −1.78653376280347642819595150783, −1.30938214809253349820735118245, 0,
1.30938214809253349820735118245, 1.78653376280347642819595150783, 3.32423421847563609552663271984, 3.93378307595772972619775208564, 4.36290887622285205528535139990, 5.29583062804514499694552500206, 6.30880350363976726250300947195, 7.17413275748935259068827942192, 7.66817485336258764470169066222
