| L(s) = 1 | − 2.57·2-s − 3-s + 4.62·4-s + 0.687·5-s + 2.57·6-s + 7-s − 6.74·8-s + 9-s − 1.76·10-s − 3.36·11-s − 4.62·12-s + 6.07·13-s − 2.57·14-s − 0.687·15-s + 8.10·16-s − 5.49·17-s − 2.57·18-s + 1.62·19-s + 3.17·20-s − 21-s + 8.65·22-s − 1.78·23-s + 6.74·24-s − 4.52·25-s − 15.6·26-s − 27-s + 4.62·28-s + ⋯ |
| L(s) = 1 | − 1.81·2-s − 0.577·3-s + 2.31·4-s + 0.307·5-s + 1.05·6-s + 0.377·7-s − 2.38·8-s + 0.333·9-s − 0.559·10-s − 1.01·11-s − 1.33·12-s + 1.68·13-s − 0.687·14-s − 0.177·15-s + 2.02·16-s − 1.33·17-s − 0.606·18-s + 0.373·19-s + 0.710·20-s − 0.218·21-s + 1.84·22-s − 0.372·23-s + 1.37·24-s − 0.905·25-s − 3.06·26-s − 0.192·27-s + 0.873·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4767 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4767 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5523699908\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5523699908\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 227 | \( 1 + T \) |
| good | 2 | \( 1 + 2.57T + 2T^{2} \) |
| 5 | \( 1 - 0.687T + 5T^{2} \) |
| 11 | \( 1 + 3.36T + 11T^{2} \) |
| 13 | \( 1 - 6.07T + 13T^{2} \) |
| 17 | \( 1 + 5.49T + 17T^{2} \) |
| 19 | \( 1 - 1.62T + 19T^{2} \) |
| 23 | \( 1 + 1.78T + 23T^{2} \) |
| 29 | \( 1 + 6.70T + 29T^{2} \) |
| 31 | \( 1 - 0.373T + 31T^{2} \) |
| 37 | \( 1 + 0.687T + 37T^{2} \) |
| 41 | \( 1 - 10.7T + 41T^{2} \) |
| 43 | \( 1 + 5.83T + 43T^{2} \) |
| 47 | \( 1 - 9.85T + 47T^{2} \) |
| 53 | \( 1 + 8.46T + 53T^{2} \) |
| 59 | \( 1 + 6.25T + 59T^{2} \) |
| 61 | \( 1 + 6.53T + 61T^{2} \) |
| 67 | \( 1 + 3.91T + 67T^{2} \) |
| 71 | \( 1 - 5.03T + 71T^{2} \) |
| 73 | \( 1 - 14.2T + 73T^{2} \) |
| 79 | \( 1 + 5.34T + 79T^{2} \) |
| 83 | \( 1 + 0.277T + 83T^{2} \) |
| 89 | \( 1 - 7.06T + 89T^{2} \) |
| 97 | \( 1 - 18.1T + 97T^{2} \) |
| 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
−8.276356180835863801882355929815, −7.76069787611087631605726824895, −7.11044270004605637003444828499, −6.06506351557402642946082699420, −5.94214693976160597201091245249, −4.69244064029647216347399069653, −3.55094854826624337446541459635, −2.29666667660593744753983605406, −1.66391915682560388238027416956, −0.55896217455050025519591518639,
0.55896217455050025519591518639, 1.66391915682560388238027416956, 2.29666667660593744753983605406, 3.55094854826624337446541459635, 4.69244064029647216347399069653, 5.94214693976160597201091245249, 6.06506351557402642946082699420, 7.11044270004605637003444828499, 7.76069787611087631605726824895, 8.276356180835863801882355929815
