| L(s) = 1 | + 0.445·2-s − 1.80·4-s + 2.04·7-s − 1.69·8-s + 0.356·11-s − 5.60·13-s + 0.911·14-s + 2.85·16-s − 0.493·17-s − 2.69·19-s + 0.158·22-s + 7.65·23-s − 5·25-s − 2.49·26-s − 3.69·28-s − 4.49·29-s − 4.85·31-s + 4.65·32-s − 0.219·34-s − 7.58·37-s − 1.19·38-s + 7.52·41-s − 3.78·43-s − 0.643·44-s + 3.40·46-s − 8.31·47-s − 2.80·49-s + ⋯ |
| L(s) = 1 | + 0.314·2-s − 0.900·4-s + 0.774·7-s − 0.598·8-s + 0.107·11-s − 1.55·13-s + 0.243·14-s + 0.712·16-s − 0.119·17-s − 0.617·19-s + 0.0338·22-s + 1.59·23-s − 25-s − 0.489·26-s − 0.697·28-s − 0.834·29-s − 0.871·31-s + 0.822·32-s − 0.0377·34-s − 1.24·37-s − 0.194·38-s + 1.17·41-s − 0.576·43-s − 0.0969·44-s + 0.502·46-s − 1.21·47-s − 0.400·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1341 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1341 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 149 | \( 1 - T \) |
| good | 2 | \( 1 - 0.445T + 2T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 - 2.04T + 7T^{2} \) |
| 11 | \( 1 - 0.356T + 11T^{2} \) |
| 13 | \( 1 + 5.60T + 13T^{2} \) |
| 17 | \( 1 + 0.493T + 17T^{2} \) |
| 19 | \( 1 + 2.69T + 19T^{2} \) |
| 23 | \( 1 - 7.65T + 23T^{2} \) |
| 29 | \( 1 + 4.49T + 29T^{2} \) |
| 31 | \( 1 + 4.85T + 31T^{2} \) |
| 37 | \( 1 + 7.58T + 37T^{2} \) |
| 41 | \( 1 - 7.52T + 41T^{2} \) |
| 43 | \( 1 + 3.78T + 43T^{2} \) |
| 47 | \( 1 + 8.31T + 47T^{2} \) |
| 53 | \( 1 + 8.19T + 53T^{2} \) |
| 59 | \( 1 + 9.18T + 59T^{2} \) |
| 61 | \( 1 - 1.96T + 61T^{2} \) |
| 67 | \( 1 - 9.83T + 67T^{2} \) |
| 71 | \( 1 - 13.7T + 71T^{2} \) |
| 73 | \( 1 + 8.09T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 - 0.542T + 83T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
| 97 | \( 1 + 16.1T + 97T^{2} \) |
| 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
−9.312835152055496021597107136144, −8.410785562304166041231403182868, −7.66643984560408360452254394793, −6.79733287072167744431639109903, −5.52598659869975249120004507235, −4.96220703293971886155267678122, −4.22031581937955978062174765571, −3.10719721074647137587852589148, −1.78607527376461983249683545645, 0,
1.78607527376461983249683545645, 3.10719721074647137587852589148, 4.22031581937955978062174765571, 4.96220703293971886155267678122, 5.52598659869975249120004507235, 6.79733287072167744431639109903, 7.66643984560408360452254394793, 8.410785562304166041231403182868, 9.312835152055496021597107136144
