| L(s) = 1 | − 2-s + 3-s + 4-s − 3.29·5-s − 6-s − 8-s + 9-s + 3.29·10-s − 11-s + 12-s + 6.06·13-s − 3.29·15-s + 16-s − 6.11·17-s − 18-s + 0.0511·19-s − 3.29·20-s + 22-s − 6.75·23-s − 24-s + 5.82·25-s − 6.06·26-s + 27-s + 2.82·29-s + 3.29·30-s + 5.87·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.47·5-s − 0.408·6-s − 0.353·8-s + 0.333·9-s + 1.04·10-s − 0.301·11-s + 0.288·12-s + 1.68·13-s − 0.849·15-s + 0.250·16-s − 1.48·17-s − 0.235·18-s + 0.0117·19-s − 0.735·20-s + 0.213·22-s − 1.40·23-s − 0.204·24-s + 1.16·25-s − 1.19·26-s + 0.192·27-s + 0.525·29-s + 0.600·30-s + 1.05·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.045816767\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.045816767\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + 3.29T + 5T^{2} \) |
| 13 | \( 1 - 6.06T + 13T^{2} \) |
| 17 | \( 1 + 6.11T + 17T^{2} \) |
| 19 | \( 1 - 0.0511T + 19T^{2} \) |
| 23 | \( 1 + 6.75T + 23T^{2} \) |
| 29 | \( 1 - 2.82T + 29T^{2} \) |
| 31 | \( 1 - 5.87T + 31T^{2} \) |
| 37 | \( 1 + 8.31T + 37T^{2} \) |
| 41 | \( 1 - 6.11T + 41T^{2} \) |
| 43 | \( 1 - 2.90T + 43T^{2} \) |
| 47 | \( 1 - 1.22T + 47T^{2} \) |
| 53 | \( 1 - 3.00T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 - 7.23T + 61T^{2} \) |
| 67 | \( 1 + 1.68T + 67T^{2} \) |
| 71 | \( 1 + 6.47T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 + 9.13T + 79T^{2} \) |
| 83 | \( 1 - 0.951T + 83T^{2} \) |
| 89 | \( 1 - 16.5T + 89T^{2} \) |
| 97 | \( 1 + 14.7T + 97T^{2} \) |
| 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
−8.501135857166400664191450983079, −8.189516413735806325798664077993, −7.36582094825518765890671566167, −6.67654234188808538216326521516, −5.85267106514597354798289223264, −4.41949677115148371712792643919, −3.93618836947834020180445358727, −3.06303823879499339361045703378, −1.98493571374003736049485483123, −0.65443612567779958448135464163,
0.65443612567779958448135464163, 1.98493571374003736049485483123, 3.06303823879499339361045703378, 3.93618836947834020180445358727, 4.41949677115148371712792643919, 5.85267106514597354798289223264, 6.67654234188808538216326521516, 7.36582094825518765890671566167, 8.189516413735806325798664077993, 8.501135857166400664191450983079
