| L(s) = 1 | + (−6.22 + 7.41i)2-s + (3.87 + 10.6i)3-s + (−5.15 − 29.2i)4-s + (−26.3 + 149. i)5-s + (−102. − 37.4i)6-s + (−266. − 462. i)7-s + (−287. − 166. i)8-s + (460. − 386. i)9-s + (−943. − 1.12e3i)10-s + (−1.05e3 + 1.82e3i)11-s + (290. − 167. i)12-s + (−381. + 1.04e3i)13-s + (5.08e3 + 896. i)14-s + (−1.69e3 + 298. i)15-s + (4.80e3 − 1.74e3i)16-s + (−642. − 538. i)17-s + ⋯ |
| L(s) = 1 | + (−0.777 + 0.926i)2-s + (0.143 + 0.393i)3-s + (−0.0805 − 0.456i)4-s + (−0.210 + 1.19i)5-s + (−0.476 − 0.173i)6-s + (−0.777 − 1.34i)7-s + (−0.561 − 0.324i)8-s + (0.631 − 0.529i)9-s + (−0.943 − 1.12i)10-s + (−0.792 + 1.37i)11-s + (0.168 − 0.0971i)12-s + (−0.173 + 0.477i)13-s + (1.85 + 0.326i)14-s + (−0.500 + 0.0883i)15-s + (1.17 − 0.427i)16-s + (−0.130 − 0.109i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.896 + 0.442i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.896 + 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.128264 - 0.549286i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.128264 - 0.549286i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + (2.76e3 - 6.27e3i)T \) |
| good | 2 | \( 1 + (6.22 - 7.41i)T + (-11.1 - 63.0i)T^{2} \) |
| 3 | \( 1 + (-3.87 - 10.6i)T + (-558. + 468. i)T^{2} \) |
| 5 | \( 1 + (26.3 - 149. i)T + (-1.46e4 - 5.34e3i)T^{2} \) |
| 7 | \( 1 + (266. + 462. i)T + (-5.88e4 + 1.01e5i)T^{2} \) |
| 11 | \( 1 + (1.05e3 - 1.82e3i)T + (-8.85e5 - 1.53e6i)T^{2} \) |
| 13 | \( 1 + (381. - 1.04e3i)T + (-3.69e6 - 3.10e6i)T^{2} \) |
| 17 | \( 1 + (642. + 538. i)T + (4.19e6 + 2.37e7i)T^{2} \) |
| 23 | \( 1 + (-1.51e3 - 8.60e3i)T + (-1.39e8 + 5.06e7i)T^{2} \) |
| 29 | \( 1 + (41.4 + 49.3i)T + (-1.03e8 + 5.85e8i)T^{2} \) |
| 31 | \( 1 + (4.52e4 - 2.61e4i)T + (4.43e8 - 7.68e8i)T^{2} \) |
| 37 | \( 1 + 8.61e3iT - 2.56e9T^{2} \) |
| 41 | \( 1 + (-2.99e4 - 8.23e4i)T + (-3.63e9 + 3.05e9i)T^{2} \) |
| 43 | \( 1 + (-9.87e3 + 5.60e4i)T + (-5.94e9 - 2.16e9i)T^{2} \) |
| 47 | \( 1 + (-3.76e4 + 3.15e4i)T + (1.87e9 - 1.06e10i)T^{2} \) |
| 53 | \( 1 + (-1.74e5 + 3.07e4i)T + (2.08e10 - 7.58e9i)T^{2} \) |
| 59 | \( 1 + (4.70e4 - 5.61e4i)T + (-7.32e9 - 4.15e10i)T^{2} \) |
| 61 | \( 1 + (5.27e4 + 2.99e5i)T + (-4.84e10 + 1.76e10i)T^{2} \) |
| 67 | \( 1 + (-3.16e5 - 3.76e5i)T + (-1.57e10 + 8.90e10i)T^{2} \) |
| 71 | \( 1 + (1.03e5 + 1.82e4i)T + (1.20e11 + 4.38e10i)T^{2} \) |
| 73 | \( 1 + (-3.19e5 + 1.16e5i)T + (1.15e11 - 9.72e10i)T^{2} \) |
| 79 | \( 1 + (-1.73e5 - 4.75e5i)T + (-1.86e11 + 1.56e11i)T^{2} \) |
| 83 | \( 1 + (4.31e5 + 7.46e5i)T + (-1.63e11 + 2.83e11i)T^{2} \) |
| 89 | \( 1 + (5.19e4 - 1.42e5i)T + (-3.80e11 - 3.19e11i)T^{2} \) |
| 97 | \( 1 + (1.49e5 - 1.78e5i)T + (-1.44e11 - 8.20e11i)T^{2} \) |
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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
−17.83117861812672976425486986439, −16.55875006888061526282659908207, −15.50264122013104947683815242482, −14.56397271312176941218721319009, −12.70152783448644441435664233601, −10.43998329075635224857886262902, −9.639669142501321180133756218412, −7.37746526320400054901609540567, −6.85520608170092219297637256404, −3.69602773161898534072290343435,
0.45469680476827976340359329473, 2.50723681981252624224118481443, 5.58533833114961683923883865906, 8.345364113906680928363900900351, 9.195222374487136199383595926592, 10.84625510022264213980824277421, 12.39768003629808286724778775613, 13.09837391284587346209258717929, 15.46419110925033711637575414832, 16.52126420107553821853690622072
