| L(s) = 1 | + 3.37·2-s − 7.36·3-s + 3.37·4-s − 10.1·5-s − 24.8·6-s − 17.4·7-s − 15.6·8-s + 27.2·9-s − 34.0·10-s + 51.5·11-s − 24.8·12-s + 75.2·13-s − 58.9·14-s + 74.4·15-s − 79.6·16-s + 91.8·18-s + 28·19-s − 34.0·20-s + 128.·21-s + 173.·22-s + 19.1·23-s + 114.·24-s − 22.8·25-s + 253.·26-s − 1.72·27-s − 58.9·28-s + 70.7·29-s + ⋯ |
| L(s) = 1 | + 1.19·2-s − 1.41·3-s + 0.421·4-s − 0.903·5-s − 1.68·6-s − 0.943·7-s − 0.689·8-s + 1.00·9-s − 1.07·10-s + 1.41·11-s − 0.597·12-s + 1.60·13-s − 1.12·14-s + 1.28·15-s − 1.24·16-s + 1.20·18-s + 0.338·19-s − 0.381·20-s + 1.33·21-s + 1.68·22-s + 0.173·23-s + 0.977·24-s − 0.182·25-s + 1.91·26-s − 0.0122·27-s − 0.397·28-s + 0.452·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.368370798\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.368370798\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 - 3.37T + 8T^{2} \) |
| 3 | \( 1 + 7.36T + 27T^{2} \) |
| 5 | \( 1 + 10.1T + 125T^{2} \) |
| 7 | \( 1 + 17.4T + 343T^{2} \) |
| 11 | \( 1 - 51.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 75.2T + 2.19e3T^{2} \) |
| 19 | \( 1 - 28T + 6.85e3T^{2} \) |
| 23 | \( 1 - 19.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 70.7T + 2.43e4T^{2} \) |
| 31 | \( 1 - 41.4T + 2.97e4T^{2} \) |
| 37 | \( 1 - 135.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 288.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 88.2T + 7.95e4T^{2} \) |
| 47 | \( 1 - 157.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 120.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 696.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 683.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 123.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 225.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 919.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 354.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 955.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 617.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 428.T + 9.12e5T^{2} \) |
| 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
−11.68331603704816263125104086076, −10.99439504813055507425944708762, −9.630099623584483946673929578372, −8.493980463985487028769066174522, −6.67068583348650173544511306098, −6.35862895019950890723224605967, −5.31313732143508092667245158770, −4.07593373390113178580378211991, −3.49202146998452683787844055247, −0.74365199954680531701580993935,
0.74365199954680531701580993935, 3.49202146998452683787844055247, 4.07593373390113178580378211991, 5.31313732143508092667245158770, 6.35862895019950890723224605967, 6.67068583348650173544511306098, 8.493980463985487028769066174522, 9.630099623584483946673929578372, 10.99439504813055507425944708762, 11.68331603704816263125104086076
