| L(s) = 1 | − 0.233·2-s − 9·3-s − 31.9·4-s + 18.8·5-s + 2.09·6-s − 97.3·7-s + 14.9·8-s + 81·9-s − 4.40·10-s + 331.·11-s + 287.·12-s − 212.·13-s + 22.7·14-s − 169.·15-s + 1.01e3·16-s + 1.97e3·17-s − 18.8·18-s + 1.41e3·19-s − 603.·20-s + 876.·21-s − 77.2·22-s + 529·23-s − 134.·24-s − 2.76e3·25-s + 49.5·26-s − 729·27-s + 3.11e3·28-s + ⋯ |
| L(s) = 1 | − 0.0412·2-s − 0.577·3-s − 0.998·4-s + 0.337·5-s + 0.0237·6-s − 0.751·7-s + 0.0823·8-s + 0.333·9-s − 0.0139·10-s + 0.825·11-s + 0.576·12-s − 0.348·13-s + 0.0309·14-s − 0.195·15-s + 0.994·16-s + 1.65·17-s − 0.0137·18-s + 0.897·19-s − 0.337·20-s + 0.433·21-s − 0.0340·22-s + 0.208·23-s − 0.0475·24-s − 0.885·25-s + 0.0143·26-s − 0.192·27-s + 0.749·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.096278664\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.096278664\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 9T \) |
| 23 | \( 1 - 529T \) |
| good | 2 | \( 1 + 0.233T + 32T^{2} \) |
| 5 | \( 1 - 18.8T + 3.12e3T^{2} \) |
| 7 | \( 1 + 97.3T + 1.68e4T^{2} \) |
| 11 | \( 1 - 331.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 212.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.97e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.41e3T + 2.47e6T^{2} \) |
| 29 | \( 1 + 1.78e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 9.98e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 3.47e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 3.57e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 3.38e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.31e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.46e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.73e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 9.71e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.61e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.34e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.81e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.77e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.89e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 9.37e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.50e5T + 8.58e9T^{2} \) |
| 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
−13.71612637561532631766554243897, −12.64842566462738427834007242351, −11.71636962928173137121381413647, −9.920450504393496299288689827551, −9.575120421612229547603809582017, −7.901385799922979622912873991490, −6.29728703296672761205215211917, −5.11870551815242977273220823444, −3.55422973256952160286379555352, −0.890969353120466293456457410360,
0.890969353120466293456457410360, 3.55422973256952160286379555352, 5.11870551815242977273220823444, 6.29728703296672761205215211917, 7.901385799922979622912873991490, 9.575120421612229547603809582017, 9.920450504393496299288689827551, 11.71636962928173137121381413647, 12.64842566462738427834007242351, 13.71612637561532631766554243897
