| L(s) = 1 | + (3.32 + 5.76i)2-s + (−131. − 227. i)3-s + (233. − 405. i)4-s + (−77.5 − 134. i)5-s + (873. − 1.51e3i)6-s − 3.03e3·7-s + 6.51e3·8-s + (−2.46e4 + 4.26e4i)9-s + (515. − 893. i)10-s − 8.23e3·11-s − 1.22e5·12-s + (3.01e4 − 5.21e4i)13-s + (−1.00e4 − 1.74e4i)14-s + (−2.03e4 + 3.52e4i)15-s + (−9.80e4 − 1.69e5i)16-s + (3.65e3 + 6.32e3i)17-s + ⋯ |
| L(s) = 1 | + (0.146 + 0.254i)2-s + (−0.935 − 1.62i)3-s + (0.456 − 0.791i)4-s + (−0.0554 − 0.0961i)5-s + (0.275 − 0.476i)6-s − 0.477·7-s + 0.562·8-s + (−1.25 + 2.16i)9-s + (0.0163 − 0.0282i)10-s − 0.169·11-s − 1.70·12-s + (0.292 − 0.506i)13-s + (−0.0702 − 0.121i)14-s + (−0.103 + 0.179i)15-s + (−0.374 − 0.647i)16-s + (0.0106 + 0.0183i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.959 - 0.283i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.959 - 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.113514 + 0.785349i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.113514 + 0.785349i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + (3.19e5 - 4.69e5i)T \) |
| good | 2 | \( 1 + (-3.32 - 5.76i)T + (-256 + 443. i)T^{2} \) |
| 3 | \( 1 + (131. + 227. i)T + (-9.84e3 + 1.70e4i)T^{2} \) |
| 5 | \( 1 + (77.5 + 134. i)T + (-9.76e5 + 1.69e6i)T^{2} \) |
| 7 | \( 1 + 3.03e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 8.23e3T + 2.35e9T^{2} \) |
| 13 | \( 1 + (-3.01e4 + 5.21e4i)T + (-5.30e9 - 9.18e9i)T^{2} \) |
| 17 | \( 1 + (-3.65e3 - 6.32e3i)T + (-5.92e10 + 1.02e11i)T^{2} \) |
| 23 | \( 1 + (8.97e5 - 1.55e6i)T + (-9.00e11 - 1.55e12i)T^{2} \) |
| 29 | \( 1 + (-3.22e6 + 5.59e6i)T + (-7.25e12 - 1.25e13i)T^{2} \) |
| 31 | \( 1 + 8.52e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 3.89e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + (7.18e6 + 1.24e7i)T + (-1.63e14 + 2.83e14i)T^{2} \) |
| 43 | \( 1 + (1.07e7 + 1.86e7i)T + (-2.51e14 + 4.35e14i)T^{2} \) |
| 47 | \( 1 + (-1.20e7 + 2.09e7i)T + (-5.59e14 - 9.69e14i)T^{2} \) |
| 53 | \( 1 + (3.12e7 - 5.41e7i)T + (-1.64e15 - 2.85e15i)T^{2} \) |
| 59 | \( 1 + (4.03e7 + 6.99e7i)T + (-4.33e15 + 7.50e15i)T^{2} \) |
| 61 | \( 1 + (-7.77e7 + 1.34e8i)T + (-5.84e15 - 1.01e16i)T^{2} \) |
| 67 | \( 1 + (9.84e7 - 1.70e8i)T + (-1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 + (-4.29e7 - 7.43e7i)T + (-2.29e16 + 3.97e16i)T^{2} \) |
| 73 | \( 1 + (9.37e7 + 1.62e8i)T + (-2.94e16 + 5.09e16i)T^{2} \) |
| 79 | \( 1 + (-2.10e8 - 3.64e8i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 - 2.50e7T + 1.86e17T^{2} \) |
| 89 | \( 1 + (-4.65e8 + 8.05e8i)T + (-1.75e17 - 3.03e17i)T^{2} \) |
| 97 | \( 1 + (-4.18e8 - 7.25e8i)T + (-3.80e17 + 6.58e17i)T^{2} \) |
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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
−15.82627750452006147009296387262, −14.05666142302846016020717339352, −12.91477010620395747302836113812, −11.73433103800588592149413834762, −10.43152600753872464408813991615, −7.81495078155235001790324971953, −6.49445031064131494593056833956, −5.58378262998705292878041954123, −1.89338289435874851822627367112, −0.38644760580705369382800888892,
3.29252239885725759944157340442, 4.64758719364259103901930453070, 6.53750554175202214724610039065, 8.978254419254704265408077729039, 10.53765458462290896446264284161, 11.37632705761590030428977476802, 12.69774325937769170462943863714, 14.84322137277971638873110169878, 16.21333333399322198116262480320, 16.53108404313968236935500768726
