| L(s) = 1 | + 43.3·2-s + 27.4·3-s + 1.37e3·4-s + 1.23e3·5-s + 1.18e3·6-s − 1.06e4·7-s + 3.72e4·8-s − 1.89e4·9-s + 5.36e4·10-s + 6.69e4·11-s + 3.75e4·12-s − 2.27e4·13-s − 4.64e5·14-s + 3.38e4·15-s + 9.16e5·16-s − 3.40e5·17-s − 8.21e5·18-s + 1.30e5·19-s + 1.69e6·20-s − 2.93e5·21-s + 2.90e6·22-s − 1.01e6·23-s + 1.02e6·24-s − 4.24e5·25-s − 9.85e5·26-s − 1.05e6·27-s − 1.46e7·28-s + ⋯ |
| L(s) = 1 | + 1.91·2-s + 0.195·3-s + 2.67·4-s + 0.884·5-s + 0.374·6-s − 1.68·7-s + 3.21·8-s − 0.961·9-s + 1.69·10-s + 1.37·11-s + 0.523·12-s − 0.220·13-s − 3.22·14-s + 0.172·15-s + 3.49·16-s − 0.989·17-s − 1.84·18-s + 0.229·19-s + 2.36·20-s − 0.328·21-s + 2.64·22-s − 0.759·23-s + 0.628·24-s − 0.217·25-s − 0.423·26-s − 0.383·27-s − 4.50·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(5.341499086\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.341499086\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 - 1.30e5T \) |
| good | 2 | \( 1 - 43.3T + 512T^{2} \) |
| 3 | \( 1 - 27.4T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.23e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 1.06e4T + 4.03e7T^{2} \) |
| 11 | \( 1 - 6.69e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 2.27e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 3.40e5T + 1.18e11T^{2} \) |
| 23 | \( 1 + 1.01e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.04e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 1.69e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.45e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.53e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.25e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + 2.16e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 1.05e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 5.88e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.92e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.17e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 3.49e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.32e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 2.58e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 4.92e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 6.00e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 6.79e8T + 7.60e17T^{2} \) |
| 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
−16.00513554667423621160728839371, −14.58852855286389171562239880101, −13.69930692407401268054762884623, −12.76257983409799674762636351599, −11.46021366921676606866451032220, −9.592617374467148539667663564559, −6.62357282519856290156939118643, −5.86644379450180981655201839797, −3.78694178789874056213798447836, −2.42044760528025127860201510800,
2.42044760528025127860201510800, 3.78694178789874056213798447836, 5.86644379450180981655201839797, 6.62357282519856290156939118643, 9.592617374467148539667663564559, 11.46021366921676606866451032220, 12.76257983409799674762636351599, 13.69930692407401268054762884623, 14.58852855286389171562239880101, 16.00513554667423621160728839371
