| L(s) = 1 | − 2.77·2-s − 2.09·3-s + 5.72·4-s + 0.541·5-s + 5.81·6-s + 4.43·7-s − 10.3·8-s + 1.36·9-s − 1.50·10-s + 2.88·11-s − 11.9·12-s + 1.35·13-s − 12.3·14-s − 1.13·15-s + 17.3·16-s − 3.13·17-s − 3.80·18-s − 19-s + 3.10·20-s − 9.26·21-s − 8.01·22-s + 4.70·23-s + 21.6·24-s − 4.70·25-s − 3.76·26-s + 3.41·27-s + 25.3·28-s + ⋯ |
| L(s) = 1 | − 1.96·2-s − 1.20·3-s + 2.86·4-s + 0.242·5-s + 2.37·6-s + 1.67·7-s − 3.66·8-s + 0.456·9-s − 0.475·10-s + 0.869·11-s − 3.45·12-s + 0.375·13-s − 3.29·14-s − 0.292·15-s + 4.33·16-s − 0.760·17-s − 0.896·18-s − 0.229·19-s + 0.693·20-s − 2.02·21-s − 1.70·22-s + 0.980·23-s + 4.42·24-s − 0.941·25-s − 0.738·26-s + 0.656·27-s + 4.79·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5784187563\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5784187563\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + T \) |
| 317 | \( 1 - T \) |
| good | 2 | \( 1 + 2.77T + 2T^{2} \) |
| 3 | \( 1 + 2.09T + 3T^{2} \) |
| 5 | \( 1 - 0.541T + 5T^{2} \) |
| 7 | \( 1 - 4.43T + 7T^{2} \) |
| 11 | \( 1 - 2.88T + 11T^{2} \) |
| 13 | \( 1 - 1.35T + 13T^{2} \) |
| 17 | \( 1 + 3.13T + 17T^{2} \) |
| 23 | \( 1 - 4.70T + 23T^{2} \) |
| 29 | \( 1 + 3.93T + 29T^{2} \) |
| 31 | \( 1 + 5.35T + 31T^{2} \) |
| 37 | \( 1 - 2.32T + 37T^{2} \) |
| 41 | \( 1 + 12.6T + 41T^{2} \) |
| 43 | \( 1 + 0.767T + 43T^{2} \) |
| 47 | \( 1 + 3.77T + 47T^{2} \) |
| 53 | \( 1 + 11.3T + 53T^{2} \) |
| 59 | \( 1 - 2.59T + 59T^{2} \) |
| 61 | \( 1 + 10.8T + 61T^{2} \) |
| 67 | \( 1 - 13.7T + 67T^{2} \) |
| 71 | \( 1 - 9.47T + 71T^{2} \) |
| 73 | \( 1 - 9.42T + 73T^{2} \) |
| 79 | \( 1 + 6.30T + 79T^{2} \) |
| 83 | \( 1 + 12.5T + 83T^{2} \) |
| 89 | \( 1 - 8.92T + 89T^{2} \) |
| 97 | \( 1 + 3.26T + 97T^{2} \) |
| 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
−8.245629341597781812807073805137, −7.46779574686359161549561630035, −6.79028087806360395158414201748, −6.22751750318078860422084817096, −5.49833744572552427053002913873, −4.73017900427709406101901951673, −3.42539771546524671399773100536, −1.96434744678177352948880102822, −1.61704251841564484575739658632, −0.59558111908813029686746062313,
0.59558111908813029686746062313, 1.61704251841564484575739658632, 1.96434744678177352948880102822, 3.42539771546524671399773100536, 4.73017900427709406101901951673, 5.49833744572552427053002913873, 6.22751750318078860422084817096, 6.79028087806360395158414201748, 7.46779574686359161549561630035, 8.245629341597781812807073805137
