| L(s) = 1 | + (7.76 − 13.9i)2-s + (−68.3 − 22.1i)3-s + (−135. − 217. i)4-s + (19.4 − 624. i)5-s + (−840. + 783. i)6-s + 53.7i·7-s + (−4.09e3 + 207. i)8-s + (−1.13e3 − 824. i)9-s + (−8.58e3 − 5.12e3i)10-s + (−1.64e4 − 2.26e4i)11-s + (4.42e3 + 1.78e4i)12-s + (2.15e4 + 1.56e4i)13-s + (752. + 417. i)14-s + (−1.51e4 + 4.22e4i)15-s + (−2.88e4 + 5.88e4i)16-s + (−3.68e4 − 1.13e5i)17-s + ⋯ |
| L(s) = 1 | + (0.485 − 0.874i)2-s + (−0.843 − 0.274i)3-s + (−0.528 − 0.848i)4-s + (0.0311 − 0.999i)5-s + (−0.648 + 0.604i)6-s + 0.0224i·7-s + (−0.998 + 0.0505i)8-s + (−0.172 − 0.125i)9-s + (−0.858 − 0.512i)10-s + (−1.12 − 1.54i)11-s + (0.213 + 0.860i)12-s + (0.755 + 0.548i)13-s + (0.0195 + 0.0108i)14-s + (−0.300 + 0.834i)15-s + (−0.440 + 0.897i)16-s + (−0.441 − 1.35i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.593 - 0.804i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.593 - 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.592586 + 0.299110i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.592586 + 0.299110i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-7.76 + 13.9i)T \) |
| 5 | \( 1 + (-19.4 + 624. i)T \) |
| good | 3 | \( 1 + (68.3 + 22.1i)T + (5.30e3 + 3.85e3i)T^{2} \) |
| 7 | \( 1 - 53.7iT - 5.76e6T^{2} \) |
| 11 | \( 1 + (1.64e4 + 2.26e4i)T + (-6.62e7 + 2.03e8i)T^{2} \) |
| 13 | \( 1 + (-2.15e4 - 1.56e4i)T + (2.52e8 + 7.75e8i)T^{2} \) |
| 17 | \( 1 + (3.68e4 + 1.13e5i)T + (-5.64e9 + 4.10e9i)T^{2} \) |
| 19 | \( 1 + (1.01e5 - 3.28e4i)T + (1.37e10 - 9.98e9i)T^{2} \) |
| 23 | \( 1 + (-672. - 926. i)T + (-2.41e10 + 7.44e10i)T^{2} \) |
| 29 | \( 1 + (-3.89e5 + 1.19e6i)T + (-4.04e11 - 2.94e11i)T^{2} \) |
| 31 | \( 1 + (-2.21e5 + 7.19e4i)T + (6.90e11 - 5.01e11i)T^{2} \) |
| 37 | \( 1 + (-1.71e5 - 1.24e5i)T + (1.08e12 + 3.34e12i)T^{2} \) |
| 41 | \( 1 + (-2.47e6 - 1.79e6i)T + (2.46e12 + 7.59e12i)T^{2} \) |
| 43 | \( 1 + 1.78e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + (-6.12e6 - 1.99e6i)T + (1.92e13 + 1.39e13i)T^{2} \) |
| 53 | \( 1 + (1.70e6 - 5.25e6i)T + (-5.03e13 - 3.65e13i)T^{2} \) |
| 59 | \( 1 + (-9.93e6 + 1.36e7i)T + (-4.53e13 - 1.39e14i)T^{2} \) |
| 61 | \( 1 + (1.48e7 - 1.07e7i)T + (5.92e13 - 1.82e14i)T^{2} \) |
| 67 | \( 1 + (-1.35e7 + 4.40e6i)T + (3.28e14 - 2.38e14i)T^{2} \) |
| 71 | \( 1 + (1.90e7 + 6.19e6i)T + (5.22e14 + 3.79e14i)T^{2} \) |
| 73 | \( 1 + (-1.54e7 + 1.11e7i)T + (2.49e14 - 7.66e14i)T^{2} \) |
| 79 | \( 1 + (2.24e7 + 7.29e6i)T + (1.22e15 + 8.91e14i)T^{2} \) |
| 83 | \( 1 + (-2.14e6 + 6.97e5i)T + (1.82e15 - 1.32e15i)T^{2} \) |
| 89 | \( 1 + (4.63e7 - 3.36e7i)T + (1.21e15 - 3.74e15i)T^{2} \) |
| 97 | \( 1 + (3.92e7 - 1.20e8i)T + (-6.34e15 - 4.60e15i)T^{2} \) |
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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
−11.47191345287523699811157490905, −10.75402460669514881499192127515, −9.266244894362909433552899878115, −8.331900488394156988719638404288, −6.16937224345920475400717708067, −5.48897257437025065741619524683, −4.27711308397881000502120933107, −2.64670431435384187000302419536, −0.892481171927766301090152744790, −0.23096121958106715741768061603,
2.54360226908123929086266390683, 4.13743400372157774655519871898, 5.33912146188930220226787994977, 6.31500774037749802072840109396, 7.33707492938473764967612609226, 8.508495776492353939609290459081, 10.30735897199502929467091827084, 10.87935771553302185603951289588, 12.34634557465013456482227951516, 13.13599465868343050954059277407
