| L(s) = 1 | − 0.387·2-s + 1.84·3-s − 1.84·4-s + 2.41·5-s − 0.714·6-s + 4.70·7-s + 1.49·8-s + 0.394·9-s − 0.938·10-s − 2.13·11-s − 3.40·12-s + 0.173·13-s − 1.82·14-s + 4.45·15-s + 3.12·16-s − 17-s − 0.152·18-s + 4.95·19-s − 4.47·20-s + 8.66·21-s + 0.829·22-s − 0.320·23-s + 2.75·24-s + 0.848·25-s − 0.0674·26-s − 4.80·27-s − 8.69·28-s + ⋯ |
| L(s) = 1 | − 0.274·2-s + 1.06·3-s − 0.924·4-s + 1.08·5-s − 0.291·6-s + 1.77·7-s + 0.527·8-s + 0.131·9-s − 0.296·10-s − 0.644·11-s − 0.983·12-s + 0.0482·13-s − 0.487·14-s + 1.15·15-s + 0.780·16-s − 0.242·17-s − 0.0360·18-s + 1.13·19-s − 1.00·20-s + 1.89·21-s + 0.176·22-s − 0.0668·23-s + 0.561·24-s + 0.169·25-s − 0.0132·26-s − 0.923·27-s − 1.64·28-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
| \(L(1)\) |
\(\approx\) |
\(2.105621891\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.105621891\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.387T + 2T^{2} \) |
| 3 | \( 1 - 1.84T + 3T^{2} \) |
| 5 | \( 1 - 2.41T + 5T^{2} \) |
| 7 | \( 1 - 4.70T + 7T^{2} \) |
| 11 | \( 1 + 2.13T + 11T^{2} \) |
| 13 | \( 1 - 0.173T + 13T^{2} \) |
| 19 | \( 1 - 4.95T + 19T^{2} \) |
| 23 | \( 1 + 0.320T + 23T^{2} \) |
| 29 | \( 1 + 1.98T + 29T^{2} \) |
| 31 | \( 1 - 2.81T + 31T^{2} \) |
| 37 | \( 1 + 6.51T + 37T^{2} \) |
| 41 | \( 1 + 5.67T + 41T^{2} \) |
| 47 | \( 1 - 5.35T + 47T^{2} \) |
| 53 | \( 1 - 5.70T + 53T^{2} \) |
| 59 | \( 1 - 1.85T + 59T^{2} \) |
| 61 | \( 1 - 4.32T + 61T^{2} \) |
| 67 | \( 1 - 6.33T + 67T^{2} \) |
| 71 | \( 1 + 0.438T + 71T^{2} \) |
| 73 | \( 1 - 11.9T + 73T^{2} \) |
| 79 | \( 1 + 12.7T + 79T^{2} \) |
| 83 | \( 1 - 4.46T + 83T^{2} \) |
| 89 | \( 1 + 6.83T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
| 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
−10.11782239223534995132892413567, −9.413997293264321264689506885033, −8.587767827213709529277032399574, −8.146429356093794460363591899494, −7.31852538645005596443039842090, −5.52948897309245083121202755701, −5.10638391960133872539803810643, −3.88449274224380960857443379794, −2.46840009351088788128321269967, −1.46781501366470904172236697227,
1.46781501366470904172236697227, 2.46840009351088788128321269967, 3.88449274224380960857443379794, 5.10638391960133872539803810643, 5.52948897309245083121202755701, 7.31852538645005596443039842090, 8.146429356093794460363591899494, 8.587767827213709529277032399574, 9.413997293264321264689506885033, 10.11782239223534995132892413567
