| L(s) = 1 | − 0.449·2-s − 2.74·3-s − 1.79·4-s + 2.80·5-s + 1.23·6-s − 4.11·7-s + 1.70·8-s + 4.52·9-s − 1.26·10-s + 11-s + 4.93·12-s − 4.01·13-s + 1.84·14-s − 7.69·15-s + 2.82·16-s + 17-s − 2.03·18-s + 7.29·19-s − 5.04·20-s + 11.2·21-s − 0.449·22-s − 0.139·23-s − 4.68·24-s + 2.86·25-s + 1.80·26-s − 4.17·27-s + 7.39·28-s + ⋯ |
| L(s) = 1 | − 0.317·2-s − 1.58·3-s − 0.898·4-s + 1.25·5-s + 0.503·6-s − 1.55·7-s + 0.603·8-s + 1.50·9-s − 0.398·10-s + 0.301·11-s + 1.42·12-s − 1.11·13-s + 0.493·14-s − 1.98·15-s + 0.707·16-s + 0.242·17-s − 0.479·18-s + 1.67·19-s − 1.12·20-s + 2.46·21-s − 0.0958·22-s − 0.0290·23-s − 0.955·24-s + 0.572·25-s + 0.353·26-s − 0.803·27-s + 1.39·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6388684313\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6388684313\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 + 0.449T + 2T^{2} \) |
| 3 | \( 1 + 2.74T + 3T^{2} \) |
| 5 | \( 1 - 2.80T + 5T^{2} \) |
| 7 | \( 1 + 4.11T + 7T^{2} \) |
| 13 | \( 1 + 4.01T + 13T^{2} \) |
| 19 | \( 1 - 7.29T + 19T^{2} \) |
| 23 | \( 1 + 0.139T + 23T^{2} \) |
| 29 | \( 1 - 8.57T + 29T^{2} \) |
| 31 | \( 1 - 2.29T + 31T^{2} \) |
| 37 | \( 1 + 10.6T + 37T^{2} \) |
| 41 | \( 1 - 1.31T + 41T^{2} \) |
| 47 | \( 1 - 7.21T + 47T^{2} \) |
| 53 | \( 1 + 3.30T + 53T^{2} \) |
| 59 | \( 1 - 8.78T + 59T^{2} \) |
| 61 | \( 1 + 12.1T + 61T^{2} \) |
| 67 | \( 1 - 11.4T + 67T^{2} \) |
| 71 | \( 1 - 14.3T + 71T^{2} \) |
| 73 | \( 1 + 5.71T + 73T^{2} \) |
| 79 | \( 1 - 1.16T + 79T^{2} \) |
| 83 | \( 1 - 0.251T + 83T^{2} \) |
| 89 | \( 1 + 10.2T + 89T^{2} \) |
| 97 | \( 1 - 3.24T + 97T^{2} \) |
| 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
−7.63112849685945205289844280400, −6.83816275949479842102712961805, −6.45557845283797138977813082307, −5.56113100735063581701775785584, −5.34648031743419863462328576394, −4.60308421214670522934742495406, −3.56851626814884824616347707299, −2.65001746044862002728902649629, −1.29140250294908686535201234981, −0.51288876857473341568969339536,
0.51288876857473341568969339536, 1.29140250294908686535201234981, 2.65001746044862002728902649629, 3.56851626814884824616347707299, 4.60308421214670522934742495406, 5.34648031743419863462328576394, 5.56113100735063581701775785584, 6.45557845283797138977813082307, 6.83816275949479842102712961805, 7.63112849685945205289844280400
