| L(s) = 1 | + (2.36 − 7.28i)2-s + (−19.7 − 14.3i)3-s + (−21.5 − 15.6i)4-s + (54.0 + 14.4i)5-s + (−151. + 109. i)6-s − 209.·7-s + (33.0 − 24.0i)8-s + (108. + 333. i)9-s + (232. − 359. i)10-s + (66.6 − 205. i)11-s + (200. + 617. i)12-s + (−208. − 642. i)13-s + (−495. + 1.52e3i)14-s + (−858. − 1.05e3i)15-s + (−360. − 1.10e3i)16-s + (1.00e3 − 731. i)17-s + ⋯ |
| L(s) = 1 | + (0.418 − 1.28i)2-s + (−1.26 − 0.918i)3-s + (−0.674 − 0.489i)4-s + (0.966 + 0.257i)5-s + (−1.71 + 1.24i)6-s − 1.61·7-s + (0.182 − 0.132i)8-s + (0.446 + 1.37i)9-s + (0.736 − 1.13i)10-s + (0.165 − 0.510i)11-s + (0.402 + 1.23i)12-s + (−0.342 − 1.05i)13-s + (−0.675 + 2.07i)14-s + (−0.985 − 1.21i)15-s + (−0.351 − 1.08i)16-s + (0.844 − 0.613i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.963 - 0.266i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.963 - 0.266i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.145069 + 1.06853i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.145069 + 1.06853i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-54.0 - 14.4i)T \) |
| good | 2 | \( 1 + (-2.36 + 7.28i)T + (-25.8 - 18.8i)T^{2} \) |
| 3 | \( 1 + (19.7 + 14.3i)T + (75.0 + 231. i)T^{2} \) |
| 7 | \( 1 + 209.T + 1.68e4T^{2} \) |
| 11 | \( 1 + (-66.6 + 205. i)T + (-1.30e5 - 9.46e4i)T^{2} \) |
| 13 | \( 1 + (208. + 642. i)T + (-3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (-1.00e3 + 731. i)T + (4.38e5 - 1.35e6i)T^{2} \) |
| 19 | \( 1 + (923. - 670. i)T + (7.65e5 - 2.35e6i)T^{2} \) |
| 23 | \( 1 + (-425. + 1.30e3i)T + (-5.20e6 - 3.78e6i)T^{2} \) |
| 29 | \( 1 + (-4.15e3 - 3.01e3i)T + (6.33e6 + 1.95e7i)T^{2} \) |
| 31 | \( 1 + (-3.66e3 + 2.65e3i)T + (8.84e6 - 2.72e7i)T^{2} \) |
| 37 | \( 1 + (-1.02e3 - 3.14e3i)T + (-5.61e7 + 4.07e7i)T^{2} \) |
| 41 | \( 1 + (4.52e3 + 1.39e4i)T + (-9.37e7 + 6.80e7i)T^{2} \) |
| 43 | \( 1 - 8.48e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-1.23e4 - 8.95e3i)T + (7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (1.11e4 + 8.09e3i)T + (1.29e8 + 3.97e8i)T^{2} \) |
| 59 | \( 1 + (2.26e3 + 6.96e3i)T + (-5.78e8 + 4.20e8i)T^{2} \) |
| 61 | \( 1 + (-4.23e3 + 1.30e4i)T + (-6.83e8 - 4.96e8i)T^{2} \) |
| 67 | \( 1 + (3.65e4 - 2.65e4i)T + (4.17e8 - 1.28e9i)T^{2} \) |
| 71 | \( 1 + (-2.96e4 - 2.15e4i)T + (5.57e8 + 1.71e9i)T^{2} \) |
| 73 | \( 1 + (1.86e4 - 5.73e4i)T + (-1.67e9 - 1.21e9i)T^{2} \) |
| 79 | \( 1 + (-2.45e4 - 1.78e4i)T + (9.50e8 + 2.92e9i)T^{2} \) |
| 83 | \( 1 + (-2.06e4 + 1.50e4i)T + (1.21e9 - 3.74e9i)T^{2} \) |
| 89 | \( 1 + (1.86e4 - 5.74e4i)T + (-4.51e9 - 3.28e9i)T^{2} \) |
| 97 | \( 1 + (7.07e4 + 5.14e4i)T + (2.65e9 + 8.16e9i)T^{2} \) |
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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
−16.28218206911555226431886127980, −13.80955676372285697050420682438, −12.78192814844332178714945654391, −12.28547001989070964840873627179, −10.74097955029733948994742262173, −9.886079670899574647022300381437, −6.82796826685202265470248435932, −5.65267228387209714404181010402, −2.86981779760439562213787083262, −0.75843443487369434260660469116,
4.55797576916989274402307878177, 5.94558711452543048309954118988, 6.64568944581713122081671527156, 9.397375217999352326362839525571, 10.38316267195294552433317091030, 12.30775959171344213101042123125, 13.65181522045374665679622448981, 15.14017474876846113447053427114, 16.23508712450905062800713856847, 16.78600902530197082130707908590
