| L(s) = 1 | + 11.7·2-s − 53.8·3-s + 10.3·4-s − 75.2·5-s − 633.·6-s − 846.·7-s − 1.38e3·8-s + 716.·9-s − 885.·10-s + 3.59e3·11-s − 559.·12-s + 955.·13-s − 9.96e3·14-s + 4.05e3·15-s − 1.76e4·16-s − 6.53e3·17-s + 8.42e3·18-s + 6.85e3·19-s − 781.·20-s + 4.56e4·21-s + 4.22e4·22-s + 4.18e4·23-s + 7.45e4·24-s − 7.24e4·25-s + 1.12e4·26-s + 7.92e4·27-s − 8.78e3·28-s + ⋯ |
| L(s) = 1 | + 1.03·2-s − 1.15·3-s + 0.0810·4-s − 0.269·5-s − 1.19·6-s − 0.933·7-s − 0.955·8-s + 0.327·9-s − 0.280·10-s + 0.813·11-s − 0.0934·12-s + 0.120·13-s − 0.970·14-s + 0.310·15-s − 1.07·16-s − 0.322·17-s + 0.340·18-s + 0.229·19-s − 0.0218·20-s + 1.07·21-s + 0.845·22-s + 0.717·23-s + 1.10·24-s − 0.927·25-s + 0.125·26-s + 0.774·27-s − 0.0756·28-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
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 - 11.7T + 128T^{2} \) |
| 3 | \( 1 + 53.8T + 2.18e3T^{2} \) |
| 5 | \( 1 + 75.2T + 7.81e4T^{2} \) |
| 7 | \( 1 + 846.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 3.59e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 955.T + 6.27e7T^{2} \) |
| 17 | \( 1 + 6.53e3T + 4.10e8T^{2} \) |
| 23 | \( 1 - 4.18e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.70e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.53e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.95e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + 1.36e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 6.76e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 8.34e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 5.86e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.33e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 9.59e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 4.22e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 4.84e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.45e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 4.87e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.35e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 3.19e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.26e7T + 8.07e13T^{2} \) |
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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
−16.21331024645238494586283147351, −14.86712546780431632366807330641, −13.36153272833391726159767536940, −12.27853989033566855699439557329, −11.24194390156311109029488627525, −9.305010816885172165202191815402, −6.61730972335786549023836858023, −5.42800639221185321587342495500, −3.71037952877483972611751057178, 0,
3.71037952877483972611751057178, 5.42800639221185321587342495500, 6.61730972335786549023836858023, 9.305010816885172165202191815402, 11.24194390156311109029488627525, 12.27853989033566855699439557329, 13.36153272833391726159767536940, 14.86712546780431632366807330641, 16.21331024645238494586283147351
