| L(s) = 1 | − 2.29·2-s − 2.37·3-s + 3.28·4-s − 5-s + 5.46·6-s − 1.14·7-s − 2.95·8-s + 2.65·9-s + 2.29·10-s − 1.72·11-s − 7.81·12-s − 5.94·13-s + 2.63·14-s + 2.37·15-s + 0.223·16-s − 3.85·17-s − 6.10·18-s + 0.720·19-s − 3.28·20-s + 2.72·21-s + 3.96·22-s − 1.65·23-s + 7.02·24-s + 25-s + 13.6·26-s + 0.815·27-s − 3.76·28-s + ⋯ |
| L(s) = 1 | − 1.62·2-s − 1.37·3-s + 1.64·4-s − 0.447·5-s + 2.23·6-s − 0.432·7-s − 1.04·8-s + 0.885·9-s + 0.727·10-s − 0.520·11-s − 2.25·12-s − 1.64·13-s + 0.703·14-s + 0.614·15-s + 0.0559·16-s − 0.934·17-s − 1.43·18-s + 0.165·19-s − 0.734·20-s + 0.594·21-s + 0.845·22-s − 0.345·23-s + 1.43·24-s + 0.200·25-s + 2.68·26-s + 0.156·27-s − 0.711·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.002879050170\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.002879050170\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 + 2.29T + 2T^{2} \) |
| 3 | \( 1 + 2.37T + 3T^{2} \) |
| 7 | \( 1 + 1.14T + 7T^{2} \) |
| 11 | \( 1 + 1.72T + 11T^{2} \) |
| 13 | \( 1 + 5.94T + 13T^{2} \) |
| 17 | \( 1 + 3.85T + 17T^{2} \) |
| 19 | \( 1 - 0.720T + 19T^{2} \) |
| 23 | \( 1 + 1.65T + 23T^{2} \) |
| 29 | \( 1 - 2.12T + 29T^{2} \) |
| 31 | \( 1 + 8.45T + 31T^{2} \) |
| 37 | \( 1 + 5.56T + 37T^{2} \) |
| 41 | \( 1 - 2.80T + 41T^{2} \) |
| 43 | \( 1 + 4.41T + 43T^{2} \) |
| 47 | \( 1 + 11.8T + 47T^{2} \) |
| 53 | \( 1 + 0.474T + 53T^{2} \) |
| 59 | \( 1 + 4.55T + 59T^{2} \) |
| 61 | \( 1 + 10.5T + 61T^{2} \) |
| 67 | \( 1 - 1.52T + 67T^{2} \) |
| 71 | \( 1 - 3.32T + 71T^{2} \) |
| 73 | \( 1 + 10.1T + 73T^{2} \) |
| 79 | \( 1 + 6.87T + 79T^{2} \) |
| 83 | \( 1 + 5.84T + 83T^{2} \) |
| 89 | \( 1 + 5.17T + 89T^{2} \) |
| 97 | \( 1 + 13.0T + 97T^{2} \) |
| 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
−9.301076270016567696523498248377, −8.411063256184689163988799229727, −7.54158474716625335443521963149, −6.99742618612705235310122043457, −6.32277796966843450225241936336, −5.23819114383017335576256911924, −4.52977806102691097488927715502, −2.92598160659467225218004435680, −1.73013068041294582428333641528, −0.04691330305462030631495046103,
0.04691330305462030631495046103, 1.73013068041294582428333641528, 2.92598160659467225218004435680, 4.52977806102691097488927715502, 5.23819114383017335576256911924, 6.32277796966843450225241936336, 6.99742618612705235310122043457, 7.54158474716625335443521963149, 8.411063256184689163988799229727, 9.301076270016567696523498248377
