L(s) = 1 | − 2.40·2-s − 0.672·3-s + 3.76·4-s − 3.90·5-s + 1.61·6-s − 3.05·7-s − 4.22·8-s − 2.54·9-s + 9.37·10-s − 11-s − 2.53·12-s − 5.54·13-s + 7.33·14-s + 2.62·15-s + 2.62·16-s − 5.45·17-s + 6.11·18-s + 5.83·19-s − 14.6·20-s + 2.05·21-s + 2.40·22-s − 7.83·23-s + 2.84·24-s + 10.2·25-s + 13.3·26-s + 3.73·27-s − 11.4·28-s + ⋯ |
L(s) = 1 | − 1.69·2-s − 0.388·3-s + 1.88·4-s − 1.74·5-s + 0.659·6-s − 1.15·7-s − 1.49·8-s − 0.849·9-s + 2.96·10-s − 0.301·11-s − 0.730·12-s − 1.53·13-s + 1.96·14-s + 0.678·15-s + 0.657·16-s − 1.32·17-s + 1.44·18-s + 1.33·19-s − 3.28·20-s + 0.448·21-s + 0.511·22-s − 1.63·23-s + 0.580·24-s + 2.04·25-s + 2.61·26-s + 0.718·27-s − 2.17·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.01443091001\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.01443091001\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
good | 2 | \( 1 + 2.40T + 2T^{2} \) |
| 3 | \( 1 + 0.672T + 3T^{2} \) |
| 5 | \( 1 + 3.90T + 5T^{2} \) |
| 7 | \( 1 + 3.05T + 7T^{2} \) |
| 13 | \( 1 + 5.54T + 13T^{2} \) |
| 17 | \( 1 + 5.45T + 17T^{2} \) |
| 19 | \( 1 - 5.83T + 19T^{2} \) |
| 23 | \( 1 + 7.83T + 23T^{2} \) |
| 29 | \( 1 + 3.45T + 29T^{2} \) |
| 31 | \( 1 + 0.504T + 31T^{2} \) |
| 37 | \( 1 + 5.34T + 37T^{2} \) |
| 41 | \( 1 + 12.1T + 41T^{2} \) |
| 43 | \( 1 + 4.71T + 43T^{2} \) |
| 47 | \( 1 - 3.26T + 47T^{2} \) |
| 53 | \( 1 - 7.66T + 53T^{2} \) |
| 59 | \( 1 - 2.01T + 59T^{2} \) |
| 67 | \( 1 - 5.54T + 67T^{2} \) |
| 71 | \( 1 - 5.61T + 71T^{2} \) |
| 73 | \( 1 - 4.97T + 73T^{2} \) |
| 79 | \( 1 - 2.08T + 79T^{2} \) |
| 83 | \( 1 + 4.20T + 83T^{2} \) |
| 89 | \( 1 + 4.09T + 89T^{2} \) |
| 97 | \( 1 - 7.15T + 97T^{2} \) |
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
−10.32918998368721772635465401594, −9.647658448125728013666387144954, −8.712228299706126945999588402282, −8.007152121332452010518270831317, −7.21384975804685963830097141917, −6.63384738946670783395753920237, −5.13764848409107539414108402385, −3.65197846396554379993869368548, −2.51885581964096700668401127009, −0.12175164879613193476003457435,
0.12175164879613193476003457435, 2.51885581964096700668401127009, 3.65197846396554379993869368548, 5.13764848409107539414108402385, 6.63384738946670783395753920237, 7.21384975804685963830097141917, 8.007152121332452010518270831317, 8.712228299706126945999588402282, 9.647658448125728013666387144954, 10.32918998368721772635465401594