| L(s) = 1 | + (−15.0 + 5.47i)2-s + (−11.5 + 65.4i)3-s + (196. − 164. i)4-s + (−2.02e3 − 1.70e3i)5-s + (−184. − 1.04e3i)6-s + (−2.30e3 − 3.99e3i)7-s + (−2.04e3 + 3.54e3i)8-s + (1.43e4 + 5.22e3i)9-s + (3.97e4 + 1.44e4i)10-s + (−8.45e3 + 1.46e4i)11-s + (8.50e3 + 1.47e4i)12-s + (−2.11e4 − 1.20e5i)13-s + (5.65e4 + 4.74e4i)14-s + (1.34e5 − 1.13e5i)15-s + (1.13e4 − 6.45e4i)16-s + (−3.11e5 + 1.13e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.664 + 0.241i)2-s + (−0.0822 + 0.466i)3-s + (0.383 − 0.321i)4-s + (−1.45 − 1.21i)5-s + (−0.0581 − 0.329i)6-s + (−0.362 − 0.628i)7-s + (−0.176 + 0.306i)8-s + (0.728 + 0.265i)9-s + (1.25 + 0.457i)10-s + (−0.174 + 0.301i)11-s + (0.118 + 0.205i)12-s + (−0.205 − 1.16i)13-s + (0.393 + 0.329i)14-s + (0.687 − 0.576i)15-s + (0.0434 − 0.246i)16-s + (−0.905 + 0.329i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.134 - 0.990i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.134 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.291295 + 0.333530i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.291295 + 0.333530i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (15.0 - 5.47i)T \) |
| 19 | \( 1 + (-5.49e5 - 1.42e5i)T \) |
| good | 3 | \( 1 + (11.5 - 65.4i)T + (-1.84e4 - 6.73e3i)T^{2} \) |
| 5 | \( 1 + (2.02e3 + 1.70e3i)T + (3.39e5 + 1.92e6i)T^{2} \) |
| 7 | \( 1 + (2.30e3 + 3.99e3i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 + (8.45e3 - 1.46e4i)T + (-1.17e9 - 2.04e9i)T^{2} \) |
| 13 | \( 1 + (2.11e4 + 1.20e5i)T + (-9.96e9 + 3.62e9i)T^{2} \) |
| 17 | \( 1 + (3.11e5 - 1.13e5i)T + (9.08e10 - 7.62e10i)T^{2} \) |
| 23 | \( 1 + (8.87e5 - 7.44e5i)T + (3.12e11 - 1.77e12i)T^{2} \) |
| 29 | \( 1 + (-2.16e6 - 7.88e5i)T + (1.11e13 + 9.32e12i)T^{2} \) |
| 31 | \( 1 + (9.38e5 + 1.62e6i)T + (-1.32e13 + 2.28e13i)T^{2} \) |
| 37 | \( 1 + 1.37e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + (4.87e6 - 2.76e7i)T + (-3.07e14 - 1.11e14i)T^{2} \) |
| 43 | \( 1 + (-8.38e6 - 7.03e6i)T + (8.72e13 + 4.94e14i)T^{2} \) |
| 47 | \( 1 + (-4.71e7 - 1.71e7i)T + (8.57e14 + 7.19e14i)T^{2} \) |
| 53 | \( 1 + (1.49e7 - 1.25e7i)T + (5.72e14 - 3.24e15i)T^{2} \) |
| 59 | \( 1 + (-1.16e8 + 4.22e7i)T + (6.63e15 - 5.56e15i)T^{2} \) |
| 61 | \( 1 + (1.94e7 - 1.63e7i)T + (2.03e15 - 1.15e16i)T^{2} \) |
| 67 | \( 1 + (-1.31e8 - 4.78e7i)T + (2.08e16 + 1.74e16i)T^{2} \) |
| 71 | \( 1 + (1.17e8 + 9.81e7i)T + (7.96e15 + 4.51e16i)T^{2} \) |
| 73 | \( 1 + (-4.22e7 + 2.39e8i)T + (-5.53e16 - 2.01e16i)T^{2} \) |
| 79 | \( 1 + (5.90e6 - 3.34e7i)T + (-1.12e17 - 4.09e16i)T^{2} \) |
| 83 | \( 1 + (-3.44e8 - 5.96e8i)T + (-9.34e16 + 1.61e17i)T^{2} \) |
| 89 | \( 1 + (-1.29e8 - 7.36e8i)T + (-3.29e17 + 1.19e17i)T^{2} \) |
| 97 | \( 1 + (9.50e7 - 3.46e7i)T + (5.82e17 - 4.88e17i)T^{2} \) |
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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
−15.31757346308778114147511718939, −13.21337529666379820299361346211, −12.14283869752477227476795670464, −10.75778244302990389100033510793, −9.595147833596711355858493904871, −8.174257660410083108184441559949, −7.33018335096566164527005813993, −5.05479799852621521505490437105, −3.81513485040426586383205170166, −1.00215835114943494888206204087,
0.25956840329670565819359760398, 2.42553242641507288106982138038, 3.90941739838525979038341998770, 6.68953704224190638535615846223, 7.35387451795831029963354687921, 8.834920684258124640584369823675, 10.39443650090463155245616240120, 11.63723912340467652732928389488, 12.22377175503873124632067676188, 14.04396810480897491520681153507
