| L(s) = 1 | − 1.33·2-s + 8.27·3-s − 6.21·4-s + 12.3·5-s − 11.0·6-s + 31.4·7-s + 18.9·8-s + 41.4·9-s − 16.4·10-s − 11·11-s − 51.4·12-s − 82.5·13-s − 42.0·14-s + 101.·15-s + 24.4·16-s + 56.4·17-s − 55.2·18-s − 19·19-s − 76.5·20-s + 260.·21-s + 14.6·22-s + 4.15·23-s + 156.·24-s + 26.3·25-s + 110.·26-s + 119.·27-s − 195.·28-s + ⋯ |
| L(s) = 1 | − 0.471·2-s + 1.59·3-s − 0.777·4-s + 1.10·5-s − 0.751·6-s + 1.69·7-s + 0.838·8-s + 1.53·9-s − 0.519·10-s − 0.301·11-s − 1.23·12-s − 1.76·13-s − 0.802·14-s + 1.75·15-s + 0.381·16-s + 0.805·17-s − 0.723·18-s − 0.229·19-s − 0.855·20-s + 2.70·21-s + 0.142·22-s + 0.0376·23-s + 1.33·24-s + 0.210·25-s + 0.831·26-s + 0.849·27-s − 1.32·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.660665837\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.660665837\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + 11T \) |
| 19 | \( 1 + 19T \) |
| good | 2 | \( 1 + 1.33T + 8T^{2} \) |
| 3 | \( 1 - 8.27T + 27T^{2} \) |
| 5 | \( 1 - 12.3T + 125T^{2} \) |
| 7 | \( 1 - 31.4T + 343T^{2} \) |
| 13 | \( 1 + 82.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 56.4T + 4.91e3T^{2} \) |
| 23 | \( 1 - 4.15T + 1.21e4T^{2} \) |
| 29 | \( 1 + 22.9T + 2.43e4T^{2} \) |
| 31 | \( 1 - 214.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 15.8T + 5.06e4T^{2} \) |
| 41 | \( 1 - 238.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 232.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 217.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 675.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 45.3T + 2.05e5T^{2} \) |
| 61 | \( 1 + 811.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 26.5T + 3.00e5T^{2} \) |
| 71 | \( 1 + 642.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 943.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.21e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 307.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 138.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.53e3T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.07257003464603445114650298803, −10.37938647756755361724412644799, −9.769354421857963124893359956377, −8.941482626919425429438751442835, −8.037614677222007598662000696984, −7.51271419447680062040049056255, −5.30170707830746705952462254185, −4.41712301233152325787619501224, −2.56636590331600315176852181809, −1.54537674705087091199786122453,
1.54537674705087091199786122453, 2.56636590331600315176852181809, 4.41712301233152325787619501224, 5.30170707830746705952462254185, 7.51271419447680062040049056255, 8.037614677222007598662000696984, 8.941482626919425429438751442835, 9.769354421857963124893359956377, 10.37938647756755361724412644799, 12.07257003464603445114650298803
