L(s) = 1 | + (−0.965 − 0.258i)2-s + (0.258 + 0.965i)3-s + (0.866 + 0.499i)4-s − i·6-s + (−1.48 − 2.19i)7-s + (−0.707 − 0.707i)8-s + (−0.866 + 0.499i)9-s + (0.366 − 0.633i)11-s + (−0.258 + 0.965i)12-s + (−1.03 + 1.03i)13-s + (0.866 + 2.49i)14-s + (0.500 + 0.866i)16-s + (−2.19 + 0.586i)17-s + (0.965 − 0.258i)18-s + (2.09 + 3.63i)19-s + ⋯ |
L(s) = 1 | + (−0.683 − 0.183i)2-s + (0.149 + 0.557i)3-s + (0.433 + 0.249i)4-s − 0.408i·6-s + (−0.560 − 0.827i)7-s + (−0.249 − 0.249i)8-s + (−0.288 + 0.166i)9-s + (0.110 − 0.191i)11-s + (−0.0747 + 0.278i)12-s + (−0.287 + 0.287i)13-s + (0.231 + 0.668i)14-s + (0.125 + 0.216i)16-s + (−0.531 + 0.142i)17-s + (0.227 − 0.0610i)18-s + (0.481 + 0.833i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0677 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0677 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8119709915\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8119709915\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.965 + 0.258i)T \) |
| 3 | \( 1 + (-0.258 - 0.965i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (1.48 + 2.19i)T \) |
good | 11 | \( 1 + (-0.366 + 0.633i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.03 - 1.03i)T - 13iT^{2} \) |
| 17 | \( 1 + (2.19 - 0.586i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-2.09 - 3.63i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.965 - 3.60i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 - 8.19iT - 29T^{2} \) |
| 31 | \( 1 + (-6.86 - 3.96i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.57 - 1.22i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + 2.46iT - 41T^{2} \) |
| 43 | \( 1 + (0.138 + 0.138i)T + 43iT^{2} \) |
| 47 | \( 1 + (2.84 - 10.6i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (5.08 - 1.36i)T + (45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (-0.267 + 0.464i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-10.5 + 6.09i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.03 - 3.86i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + 12.8T + 71T^{2} \) |
| 73 | \( 1 + (2.07 + 7.72i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (7.03 - 4.06i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (8.24 - 8.24i)T - 83iT^{2} \) |
| 89 | \( 1 + (-3.23 - 5.59i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.94 - 4.94i)T + 97iT^{2} \) |
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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
−10.00744340816548673403179230924, −9.479856265992487477907321221710, −8.614726586217985407247273896496, −7.74254694252627626066972661681, −6.91798780379424236625811672362, −6.03941510490157880694513311125, −4.76945012935610625118538926576, −3.74861631700029099755606260400, −2.91445089024113974155686967740, −1.33593446460724260672897268042,
0.46878211698380721753043606858, 2.19217378699601207557044800962, 2.90329746112597205272395879049, 4.50418114789493385320363838921, 5.73963477115090615666549424939, 6.44109980643972388636842819023, 7.21803311306713301883937065338, 8.151901431989347226175016088909, 8.790850509929378847273591396106, 9.642419819391661488698900876437