| L(s) = 1 | + 3.56·2-s − 3·3-s + 4.68·4-s − 5.68·5-s − 10.6·6-s + 7·7-s − 11.8·8-s + 9·9-s − 20.2·10-s − 11·11-s − 14.0·12-s − 39.3·13-s + 24.9·14-s + 17.0·15-s − 79.5·16-s − 64.2·17-s + 32.0·18-s − 82.9·19-s − 26.6·20-s − 21·21-s − 39.1·22-s − 11.5·23-s + 35.4·24-s − 92.6·25-s − 139.·26-s − 27·27-s + 32.7·28-s + ⋯ |
| L(s) = 1 | + 1.25·2-s − 0.577·3-s + 0.585·4-s − 0.508·5-s − 0.726·6-s + 0.377·7-s − 0.521·8-s + 0.333·9-s − 0.640·10-s − 0.301·11-s − 0.338·12-s − 0.838·13-s + 0.475·14-s + 0.293·15-s − 1.24·16-s − 0.916·17-s + 0.419·18-s − 1.00·19-s − 0.297·20-s − 0.218·21-s − 0.379·22-s − 0.104·23-s + 0.301·24-s − 0.741·25-s − 1.05·26-s − 0.192·27-s + 0.221·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 7 | \( 1 - 7T \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 - 3.56T + 8T^{2} \) |
| 5 | \( 1 + 5.68T + 125T^{2} \) |
| 13 | \( 1 + 39.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 64.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 82.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 11.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 128.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 101.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 188.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 198.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 231.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 285.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 202.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 103.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 382.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 259.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 352.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 490.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 141.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 192.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 423.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.09e3T + 9.12e5T^{2} \) |
| 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
−11.53343683546844396106511507234, −10.73753114699161077232099477179, −9.387139974875548595705841203945, −8.099819150655491821128774100459, −6.88076476674026600294428406867, −5.84536990693627275267716683373, −4.75627284074794700173718096068, −4.07272299011925707282477207853, −2.44340098772019329130683396864, 0,
2.44340098772019329130683396864, 4.07272299011925707282477207853, 4.75627284074794700173718096068, 5.84536990693627275267716683373, 6.88076476674026600294428406867, 8.099819150655491821128774100459, 9.387139974875548595705841203945, 10.73753114699161077232099477179, 11.53343683546844396106511507234
