| L(s) = 1 | − 3.24·2-s − 81.1·3-s − 501.·4-s − 574.·5-s + 263.·6-s + 7.68e3·7-s + 3.29e3·8-s − 1.31e4·9-s + 1.86e3·10-s + 6.05e4·11-s + 4.06e4·12-s + 3.46e4·13-s − 2.49e4·14-s + 4.66e4·15-s + 2.46e5·16-s + 2.34e5·17-s + 4.25e4·18-s + 1.30e5·19-s + 2.88e5·20-s − 6.23e5·21-s − 1.96e5·22-s + 6.05e5·23-s − 2.66e5·24-s − 1.62e6·25-s − 1.12e5·26-s + 2.65e6·27-s − 3.85e6·28-s + ⋯ |
| L(s) = 1 | − 0.143·2-s − 0.578·3-s − 0.979·4-s − 0.411·5-s + 0.0829·6-s + 1.21·7-s + 0.284·8-s − 0.665·9-s + 0.0590·10-s + 1.24·11-s + 0.566·12-s + 0.336·13-s − 0.173·14-s + 0.237·15-s + 0.938·16-s + 0.681·17-s + 0.0955·18-s + 0.229·19-s + 0.402·20-s − 0.699·21-s − 0.178·22-s + 0.451·23-s − 0.164·24-s − 0.830·25-s − 0.0482·26-s + 0.963·27-s − 1.18·28-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
| \(L(5)\) |
\(\approx\) |
\(1.083652903\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.083652903\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 + 3.24T + 512T^{2} \) |
| 3 | \( 1 + 81.1T + 1.96e4T^{2} \) |
| 5 | \( 1 + 574.T + 1.95e6T^{2} \) |
| 7 | \( 1 - 7.68e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 6.05e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 3.46e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 2.34e5T + 1.18e11T^{2} \) |
| 23 | \( 1 - 6.05e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 5.19e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 3.54e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.32e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 6.50e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.56e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 4.47e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 8.47e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 4.30e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.06e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 5.31e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 3.40e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.23e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 4.18e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 2.27e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 8.88e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 6.59e8T + 7.60e17T^{2} \) |
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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
−16.82107237845544883289352643597, −14.82213214156000724161992174025, −13.90657594215878835827343347885, −12.07153683488160811391940143968, −11.07282168734190181125663383345, −9.185827916459337600786448393884, −7.897057176841006992029325721844, −5.61574120257359910981275363064, −4.11215004203933638632913066381, −0.952445434843203143590712223714,
0.952445434843203143590712223714, 4.11215004203933638632913066381, 5.61574120257359910981275363064, 7.897057176841006992029325721844, 9.185827916459337600786448393884, 11.07282168734190181125663383345, 12.07153683488160811391940143968, 13.90657594215878835827343347885, 14.82213214156000724161992174025, 16.82107237845544883289352643597
