L(s) = 1 | − 2.37·2-s + 3-s + 3.63·4-s − 1.31·5-s − 2.37·6-s − 0.859·7-s − 3.87·8-s + 9-s + 3.12·10-s + 4.30·11-s + 3.63·12-s − 4.15·13-s + 2.03·14-s − 1.31·15-s + 1.93·16-s + 2.08·17-s − 2.37·18-s − 2.45·19-s − 4.78·20-s − 0.859·21-s − 10.2·22-s + 23-s − 3.87·24-s − 3.26·25-s + 9.86·26-s + 27-s − 3.12·28-s + ⋯ |
L(s) = 1 | − 1.67·2-s + 0.577·3-s + 1.81·4-s − 0.589·5-s − 0.968·6-s − 0.324·7-s − 1.37·8-s + 0.333·9-s + 0.989·10-s + 1.29·11-s + 1.04·12-s − 1.15·13-s + 0.545·14-s − 0.340·15-s + 0.483·16-s + 0.506·17-s − 0.559·18-s − 0.563·19-s − 1.07·20-s − 0.187·21-s − 2.18·22-s + 0.208·23-s − 0.791·24-s − 0.652·25-s + 1.93·26-s + 0.192·27-s − 0.590·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
good | 2 | \( 1 + 2.37T + 2T^{2} \) |
| 5 | \( 1 + 1.31T + 5T^{2} \) |
| 7 | \( 1 + 0.859T + 7T^{2} \) |
| 11 | \( 1 - 4.30T + 11T^{2} \) |
| 13 | \( 1 + 4.15T + 13T^{2} \) |
| 17 | \( 1 - 2.08T + 17T^{2} \) |
| 19 | \( 1 + 2.45T + 19T^{2} \) |
| 31 | \( 1 + 4.70T + 31T^{2} \) |
| 37 | \( 1 + 3.94T + 37T^{2} \) |
| 41 | \( 1 + 3.17T + 41T^{2} \) |
| 43 | \( 1 - 11.0T + 43T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 + 14.0T + 53T^{2} \) |
| 59 | \( 1 - 8.47T + 59T^{2} \) |
| 61 | \( 1 + 6.39T + 61T^{2} \) |
| 67 | \( 1 - 4.74T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 - 8.54T + 73T^{2} \) |
| 79 | \( 1 + 13.0T + 79T^{2} \) |
| 83 | \( 1 + 14.2T + 83T^{2} \) |
| 89 | \( 1 - 0.795T + 89T^{2} \) |
| 97 | \( 1 - 3.92T + 97T^{2} \) |
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
−8.906707116878849210428990540735, −8.097528670896476681481933623374, −7.38310019795180556257865664146, −6.96020473791284007231526869621, −5.94269005872411107390701131635, −4.47173938147046130968370335788, −3.54585864728428179566093507799, −2.42250102426845090066656055727, −1.40243800028650936821880551474, 0,
1.40243800028650936821880551474, 2.42250102426845090066656055727, 3.54585864728428179566093507799, 4.47173938147046130968370335788, 5.94269005872411107390701131635, 6.96020473791284007231526869621, 7.38310019795180556257865664146, 8.097528670896476681481933623374, 8.906707116878849210428990540735