| L(s) = 1 | + (−13.4 + 8.59i)2-s + (−45.3 − 109. i)3-s + (108. − 231. i)4-s + (105. − 254. i)5-s + (1.55e3 + 1.08e3i)6-s + (3.07e3 − 3.07e3i)7-s + (529. + 4.06e3i)8-s + (−5.28e3 + 5.28e3i)9-s + (763. + 4.34e3i)10-s + (6.44e3 − 1.55e4i)11-s + (−3.02e4 − 1.34e3i)12-s + (9.24e3 + 2.23e4i)13-s + (−1.50e4 + 6.79e4i)14-s − 3.26e4·15-s + (−4.20e4 − 5.02e4i)16-s − 1.52e5i·17-s + ⋯ |
| L(s) = 1 | + (−0.843 + 0.536i)2-s + (−0.559 − 1.35i)3-s + (0.423 − 0.905i)4-s + (0.168 − 0.407i)5-s + (1.19 + 0.839i)6-s + (1.28 − 1.28i)7-s + (0.129 + 0.991i)8-s + (−0.805 + 0.805i)9-s + (0.0763 + 0.434i)10-s + (0.440 − 1.06i)11-s + (−1.46 − 0.0649i)12-s + (0.323 + 0.781i)13-s + (−0.392 + 1.76i)14-s − 0.644·15-s + (−0.641 − 0.767i)16-s − 1.82i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.781 + 0.624i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.781 + 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.358200 - 1.02190i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.358200 - 1.02190i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (13.4 - 8.59i)T \) |
| good | 3 | \( 1 + (45.3 + 109. i)T + (-4.63e3 + 4.63e3i)T^{2} \) |
| 5 | \( 1 + (-105. + 254. i)T + (-2.76e5 - 2.76e5i)T^{2} \) |
| 7 | \( 1 + (-3.07e3 + 3.07e3i)T - 5.76e6iT^{2} \) |
| 11 | \( 1 + (-6.44e3 + 1.55e4i)T + (-1.51e8 - 1.51e8i)T^{2} \) |
| 13 | \( 1 + (-9.24e3 - 2.23e4i)T + (-5.76e8 + 5.76e8i)T^{2} \) |
| 17 | \( 1 + 1.52e5iT - 6.97e9T^{2} \) |
| 19 | \( 1 + (-5.48e4 + 2.27e4i)T + (1.20e10 - 1.20e10i)T^{2} \) |
| 23 | \( 1 + (7.09e4 + 7.09e4i)T + 7.83e10iT^{2} \) |
| 29 | \( 1 + (8.89e5 - 3.68e5i)T + (3.53e11 - 3.53e11i)T^{2} \) |
| 31 | \( 1 - 1.25e6iT - 8.52e11T^{2} \) |
| 37 | \( 1 + (2.72e5 - 6.58e5i)T + (-2.48e12 - 2.48e12i)T^{2} \) |
| 41 | \( 1 + (-6.68e5 + 6.68e5i)T - 7.98e12iT^{2} \) |
| 43 | \( 1 + (8.95e5 - 2.16e6i)T + (-8.26e12 - 8.26e12i)T^{2} \) |
| 47 | \( 1 - 4.42e6T + 2.38e13T^{2} \) |
| 53 | \( 1 + (-3.12e6 - 1.29e6i)T + (4.40e13 + 4.40e13i)T^{2} \) |
| 59 | \( 1 + (3.33e6 + 1.38e6i)T + (1.03e14 + 1.03e14i)T^{2} \) |
| 61 | \( 1 + (4.95e6 - 2.05e6i)T + (1.35e14 - 1.35e14i)T^{2} \) |
| 67 | \( 1 + (-4.56e6 - 1.10e7i)T + (-2.87e14 + 2.87e14i)T^{2} \) |
| 71 | \( 1 + (-1.26e7 + 1.26e7i)T - 6.45e14iT^{2} \) |
| 73 | \( 1 + (-1.07e7 + 1.07e7i)T - 8.06e14iT^{2} \) |
| 79 | \( 1 - 4.25e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + (1.38e7 - 5.73e6i)T + (1.59e15 - 1.59e15i)T^{2} \) |
| 89 | \( 1 + (6.06e7 + 6.06e7i)T + 3.93e15iT^{2} \) |
| 97 | \( 1 - 9.42e7T + 7.83e15T^{2} \) |
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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
−14.19136942562810755383239729455, −13.66504214425170475209730148438, −11.67221011452469456271802243950, −10.98400479901023420847911212370, −8.969347011454931605477511356901, −7.60456327703276620375492790327, −6.80426537680185981744777641968, −5.17737992530765325327947981845, −1.44893831327978540719399052860, −0.72645740254791294410908800872,
1.99157092949347099332171582452, 4.05549250085951217764860264935, 5.73428636352603985233887594427, 8.057244504384860312923636462205, 9.356516777568928645372926187329, 10.46762771744205692963418735448, 11.33362972270037535683736964260, 12.43505263941012177485649054902, 14.98398158553871194447091815075, 15.36016630031398286154519136423
