| L(s) = 1 | − 13.2·2-s − 51.9·3-s + 47.7·4-s − 400.·5-s + 688.·6-s − 4.24·7-s + 1.06e3·8-s + 508.·9-s + 5.31e3·10-s + 6.70e3·11-s − 2.47e3·12-s − 1.19e4·13-s + 56.2·14-s + 2.08e4·15-s − 2.02e4·16-s + 5.83e3·17-s − 6.74e3·18-s − 6.85e3·19-s − 1.91e4·20-s + 220.·21-s − 8.88e4·22-s − 4.13e4·23-s − 5.52e4·24-s + 8.24e4·25-s + 1.58e5·26-s + 8.71e4·27-s − 202.·28-s + ⋯ |
| L(s) = 1 | − 1.17·2-s − 1.11·3-s + 0.373·4-s − 1.43·5-s + 1.30·6-s − 0.00467·7-s + 0.734·8-s + 0.232·9-s + 1.67·10-s + 1.51·11-s − 0.414·12-s − 1.51·13-s + 0.00548·14-s + 1.59·15-s − 1.23·16-s + 0.288·17-s − 0.272·18-s − 0.229·19-s − 0.534·20-s + 0.00519·21-s − 1.77·22-s − 0.708·23-s − 0.815·24-s + 1.05·25-s + 1.77·26-s + 0.851·27-s − 0.00174·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.2930676836\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2930676836\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + 6.85e3T \) |
| good | 2 | \( 1 + 13.2T + 128T^{2} \) |
| 3 | \( 1 + 51.9T + 2.18e3T^{2} \) |
| 5 | \( 1 + 400.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 4.24T + 8.23e5T^{2} \) |
| 11 | \( 1 - 6.70e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.19e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 5.83e3T + 4.10e8T^{2} \) |
| 23 | \( 1 + 4.13e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.87e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 4.27e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.77e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 6.83e4T + 1.94e11T^{2} \) |
| 43 | \( 1 + 2.70e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 5.21e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.36e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.15e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.13e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 4.26e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.39e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.31e4T + 1.10e13T^{2} \) |
| 79 | \( 1 - 6.30e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 7.55e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 2.92e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 9.53e6T + 8.07e13T^{2} \) |
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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
−16.98387585371099862422339587900, −16.26139537257490693385663275582, −14.60416495264324963083995363718, −12.11784639492312353811220403171, −11.45842242987324411440205722863, −9.937444114426191219385790128272, −8.290189113153891455501511039764, −6.88988887716989593763843228802, −4.47875485222833616877157468680, −0.61016624767905620120112653374,
0.61016624767905620120112653374, 4.47875485222833616877157468680, 6.88988887716989593763843228802, 8.290189113153891455501511039764, 9.937444114426191219385790128272, 11.45842242987324411440205722863, 12.11784639492312353811220403171, 14.60416495264324963083995363718, 16.26139537257490693385663275582, 16.98387585371099862422339587900
