L(s) = 1 | − 2-s − 2.37·3-s + 4-s + 2.37·5-s + 2.37·6-s − 8-s + 2.62·9-s − 2.37·10-s − 4.37·11-s − 2.37·12-s − 5.37·13-s − 5.62·15-s + 16-s + 6.74·17-s − 2.62·18-s + 0.372·19-s + 2.37·20-s + 4.37·22-s + 2.74·23-s + 2.37·24-s + 0.627·25-s + 5.37·26-s + 0.883·27-s + 10.1·29-s + 5.62·30-s − 8.74·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.36·3-s + 0.5·4-s + 1.06·5-s + 0.968·6-s − 0.353·8-s + 0.875·9-s − 0.750·10-s − 1.31·11-s − 0.684·12-s − 1.49·13-s − 1.45·15-s + 0.250·16-s + 1.63·17-s − 0.619·18-s + 0.0854·19-s + 0.530·20-s + 0.932·22-s + 0.572·23-s + 0.484·24-s + 0.125·25-s + 1.05·26-s + 0.169·27-s + 1.87·29-s + 1.02·30-s − 1.57·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
good | 3 | \( 1 + 2.37T + 3T^{2} \) |
| 5 | \( 1 - 2.37T + 5T^{2} \) |
| 11 | \( 1 + 4.37T + 11T^{2} \) |
| 13 | \( 1 + 5.37T + 13T^{2} \) |
| 17 | \( 1 - 6.74T + 17T^{2} \) |
| 19 | \( 1 - 0.372T + 19T^{2} \) |
| 23 | \( 1 - 2.74T + 23T^{2} \) |
| 29 | \( 1 - 10.1T + 29T^{2} \) |
| 31 | \( 1 + 8.74T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 43 | \( 1 + 0.627T + 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 + 10T + 53T^{2} \) |
| 59 | \( 1 - 5.37T + 59T^{2} \) |
| 61 | \( 1 - 8.37T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 + 0.255T + 71T^{2} \) |
| 73 | \( 1 + 4.11T + 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 - 5.37T + 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
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
−7.913588687712218117356791582847, −7.33315105433078133673816585691, −6.60816967283653619660568671913, −5.66570281524977280925037679614, −5.40440521408238427809354762405, −4.74402882647823446725398486014, −3.08189354489351561666007719885, −2.29203758292391617207312838769, −1.13274480395633697769138563949, 0,
1.13274480395633697769138563949, 2.29203758292391617207312838769, 3.08189354489351561666007719885, 4.74402882647823446725398486014, 5.40440521408238427809354762405, 5.66570281524977280925037679614, 6.60816967283653619660568671913, 7.33315105433078133673816585691, 7.913588687712218117356791582847