| L(s) = 1 | − 2.74·2-s + 4.78·3-s − 0.439·4-s − 5·5-s − 13.1·6-s − 9.83·7-s + 23.2·8-s − 4.06·9-s + 13.7·10-s + 11·11-s − 2.10·12-s + 85.5·13-s + 27.0·14-s − 23.9·15-s − 60.2·16-s − 49.0·17-s + 11.1·18-s + 19·19-s + 2.19·20-s − 47.0·21-s − 30.2·22-s + 154.·23-s + 111.·24-s + 25·25-s − 235.·26-s − 148.·27-s + 4.32·28-s + ⋯ |
| L(s) = 1 | − 0.972·2-s + 0.921·3-s − 0.0549·4-s − 0.447·5-s − 0.895·6-s − 0.530·7-s + 1.02·8-s − 0.150·9-s + 0.434·10-s + 0.301·11-s − 0.0506·12-s + 1.82·13-s + 0.515·14-s − 0.412·15-s − 0.941·16-s − 0.700·17-s + 0.146·18-s + 0.229·19-s + 0.0245·20-s − 0.489·21-s − 0.293·22-s + 1.39·23-s + 0.945·24-s + 0.200·25-s − 1.77·26-s − 1.06·27-s + 0.0291·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.231129417\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.231129417\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 5T \) |
| 11 | \( 1 - 11T \) |
| 19 | \( 1 - 19T \) |
| good | 2 | \( 1 + 2.74T + 8T^{2} \) |
| 3 | \( 1 - 4.78T + 27T^{2} \) |
| 7 | \( 1 + 9.83T + 343T^{2} \) |
| 13 | \( 1 - 85.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 49.0T + 4.91e3T^{2} \) |
| 23 | \( 1 - 154.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 47.0T + 2.43e4T^{2} \) |
| 31 | \( 1 + 184.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 178.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 440.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 512.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 154.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 675.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 168.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 598.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 804.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 105.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 430.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 589.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.34e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.04e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 136.T + 9.12e5T^{2} \) |
| 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
−9.216674687274675235191638616789, −8.691572651156657137432226203589, −8.320929024253779247947278892146, −7.28464098650339782311615526197, −6.48642792578321002938164156974, −5.17952601152301793122673271622, −3.86254137705479407804193519463, −3.31271108165375941945785121512, −1.85382918391826792781290735804, −0.66188245187091920844634970765,
0.66188245187091920844634970765, 1.85382918391826792781290735804, 3.31271108165375941945785121512, 3.86254137705479407804193519463, 5.17952601152301793122673271622, 6.48642792578321002938164156974, 7.28464098650339782311615526197, 8.320929024253779247947278892146, 8.691572651156657137432226203589, 9.216674687274675235191638616789
