| L(s) = 1 | − 1.00·2-s − 2.74·3-s − 0.984·4-s − 5-s + 2.76·6-s − 1.78·7-s + 3.00·8-s + 4.53·9-s + 1.00·10-s − 5.50·11-s + 2.70·12-s + 0.207·13-s + 1.79·14-s + 2.74·15-s − 1.06·16-s + 5.58·17-s − 4.56·18-s + 1.05·19-s + 0.984·20-s + 4.89·21-s + 5.54·22-s + 5.66·23-s − 8.25·24-s + 25-s − 0.209·26-s − 4.21·27-s + 1.75·28-s + ⋯ |
| L(s) = 1 | − 0.712·2-s − 1.58·3-s − 0.492·4-s − 0.447·5-s + 1.12·6-s − 0.674·7-s + 1.06·8-s + 1.51·9-s + 0.318·10-s − 1.65·11-s + 0.780·12-s + 0.0576·13-s + 0.480·14-s + 0.708·15-s − 0.265·16-s + 1.35·17-s − 1.07·18-s + 0.242·19-s + 0.220·20-s + 1.06·21-s + 1.18·22-s + 1.18·23-s − 1.68·24-s + 0.200·25-s − 0.0410·26-s − 0.810·27-s + 0.332·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 241 | \( 1 + T \) |
| good | 2 | \( 1 + 1.00T + 2T^{2} \) |
| 3 | \( 1 + 2.74T + 3T^{2} \) |
| 7 | \( 1 + 1.78T + 7T^{2} \) |
| 11 | \( 1 + 5.50T + 11T^{2} \) |
| 13 | \( 1 - 0.207T + 13T^{2} \) |
| 17 | \( 1 - 5.58T + 17T^{2} \) |
| 19 | \( 1 - 1.05T + 19T^{2} \) |
| 23 | \( 1 - 5.66T + 23T^{2} \) |
| 29 | \( 1 + 6.52T + 29T^{2} \) |
| 31 | \( 1 - 7.75T + 31T^{2} \) |
| 37 | \( 1 - 3.26T + 37T^{2} \) |
| 41 | \( 1 - 5.02T + 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 - 7.57T + 47T^{2} \) |
| 53 | \( 1 - 9.19T + 53T^{2} \) |
| 59 | \( 1 - 0.353T + 59T^{2} \) |
| 61 | \( 1 + 2.42T + 61T^{2} \) |
| 67 | \( 1 + 2.81T + 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 - 12.8T + 73T^{2} \) |
| 79 | \( 1 + 12.9T + 79T^{2} \) |
| 83 | \( 1 + 16.8T + 83T^{2} \) |
| 89 | \( 1 + 6.49T + 89T^{2} \) |
| 97 | \( 1 - 12.2T + 97T^{2} \) |
| 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
−9.649775181586320019363263192659, −8.467449643870188201479001498656, −7.63602220457578102176702468148, −7.01331750848487855667447876888, −5.75633037705439502982587811235, −5.25409162283324015030759534237, −4.38281935127944220427376512064, −3.05505570053864139016834133914, −1.01259624383104938867360201115, 0,
1.01259624383104938867360201115, 3.05505570053864139016834133914, 4.38281935127944220427376512064, 5.25409162283324015030759534237, 5.75633037705439502982587811235, 7.01331750848487855667447876888, 7.63602220457578102176702468148, 8.467449643870188201479001498656, 9.649775181586320019363263192659
