| L(s) = 1 | − 0.276·2-s + 2.26·3-s − 1.92·4-s − 3.42·5-s − 0.626·6-s + 1.42·7-s + 1.08·8-s + 2.13·9-s + 0.945·10-s − 0.0948·11-s − 4.35·12-s + 6.11·13-s − 0.392·14-s − 7.75·15-s + 3.54·16-s + 17-s − 0.589·18-s + 5.35·19-s + 6.58·20-s + 3.21·21-s + 0.0262·22-s − 5.70·23-s + 2.45·24-s + 6.70·25-s − 1.69·26-s − 1.96·27-s − 2.73·28-s + ⋯ |
| L(s) = 1 | − 0.195·2-s + 1.30·3-s − 0.961·4-s − 1.53·5-s − 0.255·6-s + 0.537·7-s + 0.383·8-s + 0.710·9-s + 0.299·10-s − 0.0286·11-s − 1.25·12-s + 1.69·13-s − 0.104·14-s − 2.00·15-s + 0.886·16-s + 0.242·17-s − 0.138·18-s + 1.22·19-s + 1.47·20-s + 0.702·21-s + 0.00558·22-s − 1.18·23-s + 0.501·24-s + 1.34·25-s − 0.331·26-s − 0.378·27-s − 0.516·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.496387053\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.496387053\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 + 0.276T + 2T^{2} \) |
| 3 | \( 1 - 2.26T + 3T^{2} \) |
| 5 | \( 1 + 3.42T + 5T^{2} \) |
| 7 | \( 1 - 1.42T + 7T^{2} \) |
| 11 | \( 1 + 0.0948T + 11T^{2} \) |
| 13 | \( 1 - 6.11T + 13T^{2} \) |
| 19 | \( 1 - 5.35T + 19T^{2} \) |
| 23 | \( 1 + 5.70T + 23T^{2} \) |
| 29 | \( 1 - 1.52T + 29T^{2} \) |
| 31 | \( 1 - 8.17T + 31T^{2} \) |
| 37 | \( 1 - 6.07T + 37T^{2} \) |
| 41 | \( 1 - 12.1T + 41T^{2} \) |
| 47 | \( 1 - 3.39T + 47T^{2} \) |
| 53 | \( 1 - 3.28T + 53T^{2} \) |
| 59 | \( 1 + 6.46T + 59T^{2} \) |
| 61 | \( 1 - 10.5T + 61T^{2} \) |
| 67 | \( 1 + 12.5T + 67T^{2} \) |
| 71 | \( 1 - 5.67T + 71T^{2} \) |
| 73 | \( 1 + 16.2T + 73T^{2} \) |
| 79 | \( 1 + 8.78T + 79T^{2} \) |
| 83 | \( 1 - 7.35T + 83T^{2} \) |
| 89 | \( 1 + 4.66T + 89T^{2} \) |
| 97 | \( 1 - 18.6T + 97T^{2} \) |
| 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
−10.20080139019364310410353859875, −9.217924382560498668953435144476, −8.502263692847091364223925412274, −7.986060151014919953722602426299, −7.57964139633225852283944507811, −5.89624011054715518598184572342, −4.43676373714857287177388222886, −3.88945398706700593450217493307, −3.03658419474574528288659288819, −1.08053269392262110825486746873,
1.08053269392262110825486746873, 3.03658419474574528288659288819, 3.88945398706700593450217493307, 4.43676373714857287177388222886, 5.89624011054715518598184572342, 7.57964139633225852283944507811, 7.986060151014919953722602426299, 8.502263692847091364223925412274, 9.217924382560498668953435144476, 10.20080139019364310410353859875
