| L(s) = 1 | − 6.83·2-s − 260.·3-s − 465.·4-s − 1.11e3·5-s + 1.78e3·6-s − 8.60e3·7-s + 6.67e3·8-s + 4.83e4·9-s + 7.62e3·10-s − 6.26e4·11-s + 1.21e5·12-s − 5.65e4·13-s + 5.87e4·14-s + 2.90e5·15-s + 1.92e5·16-s − 3.60e5·17-s − 3.30e5·18-s + 1.30e5·19-s + 5.19e5·20-s + 2.24e6·21-s + 4.28e5·22-s − 1.71e6·23-s − 1.74e6·24-s − 7.08e5·25-s + 3.86e5·26-s − 7.47e6·27-s + 4.00e6·28-s + ⋯ |
| L(s) = 1 | − 0.301·2-s − 1.85·3-s − 0.908·4-s − 0.798·5-s + 0.561·6-s − 1.35·7-s + 0.576·8-s + 2.45·9-s + 0.241·10-s − 1.29·11-s + 1.68·12-s − 0.549·13-s + 0.408·14-s + 1.48·15-s + 0.734·16-s − 1.04·17-s − 0.741·18-s + 0.229·19-s + 0.725·20-s + 2.51·21-s + 0.389·22-s − 1.27·23-s − 1.07·24-s − 0.362·25-s + 0.165·26-s − 2.70·27-s + 1.23·28-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
| \(L(5)\) |
\(\approx\) |
\(0.01160572541\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01160572541\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 + 6.83T + 512T^{2} \) |
| 3 | \( 1 + 260.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 1.11e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 8.60e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 6.26e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 5.65e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 3.60e5T + 1.18e11T^{2} \) |
| 23 | \( 1 + 1.71e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 1.11e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 8.66e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 4.46e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.67e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.66e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 4.05e6T + 1.11e15T^{2} \) |
| 53 | \( 1 + 1.25e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 5.19e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 2.43e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.81e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.61e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.99e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 2.15e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 1.50e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 1.05e9T + 3.50e17T^{2} \) |
| 97 | \( 1 + 4.82e8T + 7.60e17T^{2} \) |
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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
−16.38354066911477016182253372013, −15.74165353403234682347845311685, −13.15039855687481696412125462994, −12.34083924899773040530305363266, −10.80512000750282576328281760846, −9.711350743307644670080845729023, −7.44578945748876555650507787765, −5.75764131181370034865569555017, −4.28319713573947044517676925415, −0.094726626200472959111459146522,
0.094726626200472959111459146522, 4.28319713573947044517676925415, 5.75764131181370034865569555017, 7.44578945748876555650507787765, 9.711350743307644670080845729023, 10.80512000750282576328281760846, 12.34083924899773040530305363266, 13.15039855687481696412125462994, 15.74165353403234682347845311685, 16.38354066911477016182253372013
