| L(s) = 1 | − 3.96·2-s + 6.71·3-s + 7.70·4-s + 18.1·5-s − 26.6·6-s − 25.8·7-s + 1.18·8-s + 18.1·9-s − 71.8·10-s − 8.13·11-s + 51.7·12-s − 4.56·13-s + 102.·14-s + 121.·15-s − 66.2·16-s − 62.5·17-s − 71.7·18-s − 19·19-s + 139.·20-s − 173.·21-s + 32.2·22-s − 52.7·23-s + 7.93·24-s + 203.·25-s + 18.0·26-s − 59.7·27-s − 198.·28-s + ⋯ |
| L(s) = 1 | − 1.40·2-s + 1.29·3-s + 0.962·4-s + 1.62·5-s − 1.81·6-s − 1.39·7-s + 0.0521·8-s + 0.670·9-s − 2.27·10-s − 0.222·11-s + 1.24·12-s − 0.0974·13-s + 1.95·14-s + 2.09·15-s − 1.03·16-s − 0.892·17-s − 0.939·18-s − 0.229·19-s + 1.56·20-s − 1.80·21-s + 0.312·22-s − 0.478·23-s + 0.0674·24-s + 1.63·25-s + 0.136·26-s − 0.425·27-s − 1.34·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8560915891\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8560915891\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + 19T \) |
| good | 2 | \( 1 + 3.96T + 8T^{2} \) |
| 3 | \( 1 - 6.71T + 27T^{2} \) |
| 5 | \( 1 - 18.1T + 125T^{2} \) |
| 7 | \( 1 + 25.8T + 343T^{2} \) |
| 11 | \( 1 + 8.13T + 1.33e3T^{2} \) |
| 13 | \( 1 + 4.56T + 2.19e3T^{2} \) |
| 17 | \( 1 + 62.5T + 4.91e3T^{2} \) |
| 23 | \( 1 + 52.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 171.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 168.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 147.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 308.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 448.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 113.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 155.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 182.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 404.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 106.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 472.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 843.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 591.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 290.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 964.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 219.T + 9.12e5T^{2} \) |
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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
−18.12460994048859083837958742391, −17.07748901208720571626188924871, −15.75455184312948749633617729254, −13.94458282194016520156643692652, −13.15925565171524995021716161471, −10.20883657387201471139500452795, −9.527134055457222237238578222502, −8.551515017393265439692682089695, −6.60536515965165763306633666862, −2.42435818975861193846168331153,
2.42435818975861193846168331153, 6.60536515965165763306633666862, 8.551515017393265439692682089695, 9.527134055457222237238578222502, 10.20883657387201471139500452795, 13.15925565171524995021716161471, 13.94458282194016520156643692652, 15.75455184312948749633617729254, 17.07748901208720571626188924871, 18.12460994048859083837958742391
