| L(s) = 1 | + (−3.06 − 2.57i)2-s + (12.9 − 4.71i)3-s + (2.77 + 15.7i)4-s + (−13.1 + 74.3i)5-s + (−51.7 − 18.8i)6-s + (39.5 + 68.4i)7-s + (32.0 − 55.4i)8-s + (−40.7 + 34.1i)9-s + (231. − 194. i)10-s + (−20.5 + 35.6i)11-s + (110. + 190. i)12-s + (1.07e3 + 390. i)13-s + (54.9 − 311. i)14-s + (180. + 1.02e3i)15-s + (−240. + 87.5i)16-s + (851. + 714. i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 − 0.454i)2-s + (0.830 − 0.302i)3-s + (0.0868 + 0.492i)4-s + (−0.234 + 1.32i)5-s + (−0.587 − 0.213i)6-s + (0.304 + 0.528i)7-s + (0.176 − 0.306i)8-s + (−0.167 + 0.140i)9-s + (0.731 − 0.613i)10-s + (−0.0512 + 0.0887i)11-s + (0.220 + 0.382i)12-s + (1.75 + 0.640i)13-s + (0.0748 − 0.424i)14-s + (0.207 + 1.17i)15-s + (−0.234 + 0.0855i)16-s + (0.714 + 0.599i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.789 - 0.613i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.789 - 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.41475 + 0.484992i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.41475 + 0.484992i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (3.06 + 2.57i)T \) |
| 19 | \( 1 + (1.15e3 + 1.07e3i)T \) |
| good | 3 | \( 1 + (-12.9 + 4.71i)T + (186. - 156. i)T^{2} \) |
| 5 | \( 1 + (13.1 - 74.3i)T + (-2.93e3 - 1.06e3i)T^{2} \) |
| 7 | \( 1 + (-39.5 - 68.4i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 + (20.5 - 35.6i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 + (-1.07e3 - 390. i)T + (2.84e5 + 2.38e5i)T^{2} \) |
| 17 | \( 1 + (-851. - 714. i)T + (2.46e5 + 1.39e6i)T^{2} \) |
| 23 | \( 1 + (479. + 2.72e3i)T + (-6.04e6 + 2.20e6i)T^{2} \) |
| 29 | \( 1 + (3.06e3 - 2.57e3i)T + (3.56e6 - 2.01e7i)T^{2} \) |
| 31 | \( 1 + (832. + 1.44e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 - 1.16e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + (7.50e3 - 2.73e3i)T + (8.87e7 - 7.44e7i)T^{2} \) |
| 43 | \( 1 + (-1.21e3 + 6.88e3i)T + (-1.38e8 - 5.02e7i)T^{2} \) |
| 47 | \( 1 + (7.97e3 - 6.69e3i)T + (3.98e7 - 2.25e8i)T^{2} \) |
| 53 | \( 1 + (2.09e3 + 1.18e4i)T + (-3.92e8 + 1.43e8i)T^{2} \) |
| 59 | \( 1 + (-1.96e4 - 1.65e4i)T + (1.24e8 + 7.04e8i)T^{2} \) |
| 61 | \( 1 + (2.28e3 + 1.29e4i)T + (-7.93e8 + 2.88e8i)T^{2} \) |
| 67 | \( 1 + (-4.24e4 + 3.56e4i)T + (2.34e8 - 1.32e9i)T^{2} \) |
| 71 | \( 1 + (-1.25e4 + 7.11e4i)T + (-1.69e9 - 6.17e8i)T^{2} \) |
| 73 | \( 1 + (-6.29e3 + 2.29e3i)T + (1.58e9 - 1.33e9i)T^{2} \) |
| 79 | \( 1 + (-5.74e4 + 2.09e4i)T + (2.35e9 - 1.97e9i)T^{2} \) |
| 83 | \( 1 + (2.21e4 + 3.82e4i)T + (-1.96e9 + 3.41e9i)T^{2} \) |
| 89 | \( 1 + (9.87e4 + 3.59e4i)T + (4.27e9 + 3.58e9i)T^{2} \) |
| 97 | \( 1 + (-1.24e5 - 1.04e5i)T + (1.49e9 + 8.45e9i)T^{2} \) |
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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
−15.15435544697854760393607701563, −14.31948123769118822109713648252, −13.09093416236898796446802205002, −11.41577965519539399492788981005, −10.65788675937271898564282140952, −8.914796656647312222946453812346, −7.976175297041790830798335738420, −6.46905293738345772977408690068, −3.49490090965374492752706352124, −2.12617930179238437052451518478,
1.02952209094675037557027401850, 3.86863443638364084141292665103, 5.71107559900037857200352026858, 7.949470300849719041834279361890, 8.591895077949110973242625592785, 9.746608861702498033796705957621, 11.35619496083528059906969296606, 13.04130013596420662926316705603, 14.10153061495183416619227593951, 15.38260415472978715236546609921
