| L(s) = 1 | − 2-s + 2.82·3-s + 4-s − 0.414·5-s − 2.82·6-s − 3·7-s − 8-s + 5.00·9-s + 0.414·10-s + 2.82·11-s + 2.82·12-s + 6.82·13-s + 3·14-s − 1.17·15-s + 16-s + 4.82·17-s − 5.00·18-s − 3.65·19-s − 0.414·20-s − 8.48·21-s − 2.82·22-s + 3.17·23-s − 2.82·24-s − 4.82·25-s − 6.82·26-s + 5.65·27-s − 3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.63·3-s + 0.5·4-s − 0.185·5-s − 1.15·6-s − 1.13·7-s − 0.353·8-s + 1.66·9-s + 0.130·10-s + 0.852·11-s + 0.816·12-s + 1.89·13-s + 0.801·14-s − 0.302·15-s + 0.250·16-s + 1.17·17-s − 1.17·18-s − 0.838·19-s − 0.0926·20-s − 1.85·21-s − 0.603·22-s + 0.661·23-s − 0.577·24-s − 0.965·25-s − 1.33·26-s + 1.08·27-s − 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.557803473\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.557803473\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 167 | \( 1 - T \) |
| good | 3 | \( 1 - 2.82T + 3T^{2} \) |
| 5 | \( 1 + 0.414T + 5T^{2} \) |
| 7 | \( 1 + 3T + 7T^{2} \) |
| 11 | \( 1 - 2.82T + 11T^{2} \) |
| 13 | \( 1 - 6.82T + 13T^{2} \) |
| 17 | \( 1 - 4.82T + 17T^{2} \) |
| 19 | \( 1 + 3.65T + 19T^{2} \) |
| 23 | \( 1 - 3.17T + 23T^{2} \) |
| 29 | \( 1 - 1.17T + 29T^{2} \) |
| 31 | \( 1 + 7.82T + 31T^{2} \) |
| 37 | \( 1 + 7.24T + 37T^{2} \) |
| 41 | \( 1 - 0.343T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 + 9.58T + 53T^{2} \) |
| 59 | \( 1 - 6.41T + 59T^{2} \) |
| 61 | \( 1 - 8.48T + 61T^{2} \) |
| 67 | \( 1 - 0.414T + 67T^{2} \) |
| 71 | \( 1 + 2.48T + 71T^{2} \) |
| 73 | \( 1 + 13.6T + 73T^{2} \) |
| 79 | \( 1 + 8.48T + 79T^{2} \) |
| 83 | \( 1 - 14.8T + 83T^{2} \) |
| 89 | \( 1 - 8.65T + 89T^{2} \) |
| 97 | \( 1 - 3.82T + 97T^{2} \) |
| 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
−11.40862124668015476417861597307, −10.27675099742077063652552279128, −9.427982635605966008212835267409, −8.804495338373476011558112604988, −8.105460500173576049875070469091, −7.01382435850729352939952586464, −6.07967418279830285609275530825, −3.72695237079035098531648530366, −3.30740996308260878574767635524, −1.62732020919446501342281036349,
1.62732020919446501342281036349, 3.30740996308260878574767635524, 3.72695237079035098531648530366, 6.07967418279830285609275530825, 7.01382435850729352939952586464, 8.105460500173576049875070469091, 8.804495338373476011558112604988, 9.427982635605966008212835267409, 10.27675099742077063652552279128, 11.40862124668015476417861597307
