| L(s) = 1 | − 2.73·2-s + 5.45·4-s − 0.936·5-s − 0.519·7-s − 9.43·8-s + 2.55·10-s + 11-s − 4.70·13-s + 1.41·14-s + 14.8·16-s − 2.69·17-s + 3.41·19-s − 5.10·20-s − 2.73·22-s + 6.97·23-s − 4.12·25-s + 12.8·26-s − 2.83·28-s + 4.18·29-s + 5.18·31-s − 21.6·32-s + 7.35·34-s + 0.486·35-s + 2.06·37-s − 9.33·38-s + 8.83·40-s + 0.173·41-s + ⋯ |
| L(s) = 1 | − 1.93·2-s + 2.72·4-s − 0.418·5-s − 0.196·7-s − 3.33·8-s + 0.808·10-s + 0.301·11-s − 1.30·13-s + 0.378·14-s + 3.71·16-s − 0.652·17-s + 0.784·19-s − 1.14·20-s − 0.582·22-s + 1.45·23-s − 0.824·25-s + 2.51·26-s − 0.535·28-s + 0.777·29-s + 0.931·31-s − 3.83·32-s + 1.26·34-s + 0.0821·35-s + 0.340·37-s − 1.51·38-s + 1.39·40-s + 0.0270·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5029059233\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5029059233\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + 2.73T + 2T^{2} \) |
| 5 | \( 1 + 0.936T + 5T^{2} \) |
| 7 | \( 1 + 0.519T + 7T^{2} \) |
| 13 | \( 1 + 4.70T + 13T^{2} \) |
| 17 | \( 1 + 2.69T + 17T^{2} \) |
| 19 | \( 1 - 3.41T + 19T^{2} \) |
| 23 | \( 1 - 6.97T + 23T^{2} \) |
| 29 | \( 1 - 4.18T + 29T^{2} \) |
| 31 | \( 1 - 5.18T + 31T^{2} \) |
| 37 | \( 1 - 2.06T + 37T^{2} \) |
| 41 | \( 1 - 0.173T + 41T^{2} \) |
| 43 | \( 1 + 2.26T + 43T^{2} \) |
| 47 | \( 1 + 0.307T + 47T^{2} \) |
| 53 | \( 1 + 1.89T + 53T^{2} \) |
| 59 | \( 1 - 3.97T + 59T^{2} \) |
| 61 | \( 1 - 4.51T + 61T^{2} \) |
| 67 | \( 1 - 3.37T + 67T^{2} \) |
| 71 | \( 1 - 2.90T + 71T^{2} \) |
| 73 | \( 1 - 9.52T + 73T^{2} \) |
| 79 | \( 1 + 2.04T + 79T^{2} \) |
| 83 | \( 1 - 14.0T + 83T^{2} \) |
| 89 | \( 1 - 7.53T + 89T^{2} \) |
| 97 | \( 1 - 16.3T + 97T^{2} \) |
| 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
−9.835365588042082156209095642011, −9.401350255571293594702481703997, −8.500514976624525038165938026004, −7.73766929716417811074080278903, −7.03918028733771430002244040826, −6.34637654179161476578322413788, −4.93394312574895959686317806507, −3.23002603768436519332108956233, −2.22023280754702886723506608444, −0.73433714241970085875408554675,
0.73433714241970085875408554675, 2.22023280754702886723506608444, 3.23002603768436519332108956233, 4.93394312574895959686317806507, 6.34637654179161476578322413788, 7.03918028733771430002244040826, 7.73766929716417811074080278903, 8.500514976624525038165938026004, 9.401350255571293594702481703997, 9.835365588042082156209095642011
