| L(s) = 1 | + 17.8·2-s − 263.·3-s − 193.·4-s + 542.·5-s − 4.71e3·6-s − 1.23e4·7-s − 1.25e4·8-s + 4.99e4·9-s + 9.69e3·10-s − 6.22e4·11-s + 5.09e4·12-s − 1.61e3·13-s − 2.20e5·14-s − 1.43e5·15-s − 1.26e5·16-s + 2.99e5·17-s + 8.92e5·18-s + 1.78e5·19-s − 1.04e5·20-s + 3.25e6·21-s − 1.11e6·22-s + 2.13e6·23-s + 3.32e6·24-s − 1.65e6·25-s − 2.88e4·26-s − 7.99e6·27-s + 2.38e6·28-s + ⋯ |
| L(s) = 1 | + 0.789·2-s − 1.88·3-s − 0.377·4-s + 0.388·5-s − 1.48·6-s − 1.94·7-s − 1.08·8-s + 2.53·9-s + 0.306·10-s − 1.28·11-s + 0.709·12-s − 0.0156·13-s − 1.53·14-s − 0.730·15-s − 0.480·16-s + 0.869·17-s + 2.00·18-s + 0.313·19-s − 0.146·20-s + 3.65·21-s − 1.01·22-s + 1.59·23-s + 2.04·24-s − 0.849·25-s − 0.0123·26-s − 2.89·27-s + 0.731·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.4941183525\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4941183525\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 - 3.41e6T \) |
| good | 2 | \( 1 - 17.8T + 512T^{2} \) |
| 3 | \( 1 + 263.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 542.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 1.23e4T + 4.03e7T^{2} \) |
| 11 | \( 1 + 6.22e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.61e3T + 1.06e10T^{2} \) |
| 17 | \( 1 - 2.99e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 1.78e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.13e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 3.53e5T + 1.45e13T^{2} \) |
| 31 | \( 1 + 3.54e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 4.68e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 3.19e7T + 3.27e14T^{2} \) |
| 47 | \( 1 + 3.67e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 2.84e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 3.96e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.10e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 8.80e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.56e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 7.55e7T + 5.88e16T^{2} \) |
| 79 | \( 1 - 5.64e7T + 1.19e17T^{2} \) |
| 83 | \( 1 - 2.06e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 3.67e7T + 3.50e17T^{2} \) |
| 97 | \( 1 - 6.35e8T + 7.60e17T^{2} \) |
| 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
−13.29232004624317476689184955333, −12.92193153621303356304824333857, −11.95426808692530109119890535952, −10.38735140171490878142991190901, −9.606323213561448261250911552423, −6.90630278208394252984826905886, −5.81798400703499766452869016000, −5.12483474035441789849333949308, −3.36347289581504064110260853062, −0.44274824259526206808443981652,
0.44274824259526206808443981652, 3.36347289581504064110260853062, 5.12483474035441789849333949308, 5.81798400703499766452869016000, 6.90630278208394252984826905886, 9.606323213561448261250911552423, 10.38735140171490878142991190901, 11.95426808692530109119890535952, 12.92193153621303356304824333857, 13.29232004624317476689184955333
