L(s) = 1 | + 2-s + 0.161·3-s + 4-s − 5-s + 0.161·6-s + 0.164·7-s + 8-s − 2.97·9-s − 10-s + 2.35·11-s + 0.161·12-s + 13-s + 0.164·14-s − 0.161·15-s + 16-s − 3.98·17-s − 2.97·18-s + 7.89·19-s − 20-s + 0.0264·21-s + 2.35·22-s − 3.21·23-s + 0.161·24-s + 25-s + 26-s − 0.964·27-s + 0.164·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.0931·3-s + 0.5·4-s − 0.447·5-s + 0.0658·6-s + 0.0620·7-s + 0.353·8-s − 0.991·9-s − 0.316·10-s + 0.708·11-s + 0.0465·12-s + 0.277·13-s + 0.0438·14-s − 0.0416·15-s + 0.250·16-s − 0.965·17-s − 0.700·18-s + 1.81·19-s − 0.223·20-s + 0.00577·21-s + 0.501·22-s − 0.670·23-s + 0.0329·24-s + 0.200·25-s + 0.196·26-s − 0.185·27-s + 0.0310·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.864056097\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.864056097\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 - T \) |
good | 3 | \( 1 - 0.161T + 3T^{2} \) |
| 7 | \( 1 - 0.164T + 7T^{2} \) |
| 11 | \( 1 - 2.35T + 11T^{2} \) |
| 17 | \( 1 + 3.98T + 17T^{2} \) |
| 19 | \( 1 - 7.89T + 19T^{2} \) |
| 23 | \( 1 + 3.21T + 23T^{2} \) |
| 29 | \( 1 - 10.4T + 29T^{2} \) |
| 37 | \( 1 - 0.957T + 37T^{2} \) |
| 41 | \( 1 - 3.80T + 41T^{2} \) |
| 43 | \( 1 + 11.7T + 43T^{2} \) |
| 47 | \( 1 - 3.52T + 47T^{2} \) |
| 53 | \( 1 + 5.93T + 53T^{2} \) |
| 59 | \( 1 - 4.00T + 59T^{2} \) |
| 61 | \( 1 - 6.23T + 61T^{2} \) |
| 67 | \( 1 + 1.88T + 67T^{2} \) |
| 71 | \( 1 - 8.81T + 71T^{2} \) |
| 73 | \( 1 - 15.5T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 - 13.7T + 83T^{2} \) |
| 89 | \( 1 - 13.9T + 89T^{2} \) |
| 97 | \( 1 - 9.61T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.251430289727680649520081299775, −7.82689981189740468994629117181, −6.64018880466484488672503892140, −6.40433113948798909888750304049, −5.29683663370449361711474933847, −4.75086824867708307750725464268, −3.73248573101481237449953252672, −3.16777570505801057732504276925, −2.20528882255948944581608683650, −0.876445763163244354104558612460,
0.876445763163244354104558612460, 2.20528882255948944581608683650, 3.16777570505801057732504276925, 3.73248573101481237449953252672, 4.75086824867708307750725464268, 5.29683663370449361711474933847, 6.40433113948798909888750304049, 6.64018880466484488672503892140, 7.82689981189740468994629117181, 8.251430289727680649520081299775