L(s) = 1 | + (−1.16 + 0.804i)2-s + (−0.540 − 2.01i)3-s + (0.706 − 1.87i)4-s + (0.161 − 2.23i)5-s + (2.25 + 1.91i)6-s + (1.67 − 2.04i)7-s + (0.683 + 2.74i)8-s + (−1.17 + 0.679i)9-s + (1.60 + 2.72i)10-s + (−2.35 + 4.08i)11-s + (−4.15 − 0.413i)12-s + (−3.68 − 3.68i)13-s + (−0.298 + 3.72i)14-s + (−4.58 + 0.879i)15-s + (−3.00 − 2.64i)16-s + (5.98 − 1.60i)17-s + ⋯ |
L(s) = 1 | + (−0.822 + 0.568i)2-s + (−0.311 − 1.16i)3-s + (0.353 − 0.935i)4-s + (0.0721 − 0.997i)5-s + (0.918 + 0.780i)6-s + (0.632 − 0.774i)7-s + (0.241 + 0.970i)8-s + (−0.392 + 0.226i)9-s + (0.507 + 0.861i)10-s + (−0.711 + 1.23i)11-s + (−1.19 − 0.119i)12-s + (−1.02 − 1.02i)13-s + (−0.0796 + 0.996i)14-s + (−1.18 + 0.227i)15-s + (−0.750 − 0.660i)16-s + (1.45 − 0.389i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.519 + 0.854i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.519 + 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.350169 - 0.622736i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.350169 - 0.622736i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.16 - 0.804i)T \) |
| 5 | \( 1 + (-0.161 + 2.23i)T \) |
| 7 | \( 1 + (-1.67 + 2.04i)T \) |
good | 3 | \( 1 + (0.540 + 2.01i)T + (-2.59 + 1.5i)T^{2} \) |
| 11 | \( 1 + (2.35 - 4.08i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (3.68 + 3.68i)T + 13iT^{2} \) |
| 17 | \( 1 + (-5.98 + 1.60i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (2.62 - 1.51i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.371 - 1.38i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 - 3.08T + 29T^{2} \) |
| 31 | \( 1 + (-0.111 - 0.0644i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.12 - 0.568i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 - 1.37T + 41T^{2} \) |
| 43 | \( 1 + (-2.05 + 2.05i)T - 43iT^{2} \) |
| 47 | \( 1 + (11.0 + 2.96i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (2.82 - 0.758i)T + (45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (-11.1 - 6.42i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.49 + 3.17i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.28 + 1.95i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 1.96iT - 71T^{2} \) |
| 73 | \( 1 + (-0.205 - 0.768i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (0.00532 + 0.00922i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.27 + 4.27i)T - 83iT^{2} \) |
| 89 | \( 1 + (-6.15 + 3.55i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-8.22 - 8.22i)T + 97iT^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.69890676839991951677803465234, −10.22483593368511785851177596557, −9.793680141225219427086818050636, −8.125563150701034832152257301583, −7.75016244554014522594120100572, −7.00003144797781665987958555648, −5.57948909683941193171617424821, −4.79155149315795994701721893913, −1.97279232213947278667461752493, −0.73995300224962442771154111254,
2.37281138467943191414013113299, 3.52886981363795947237339099125, 4.91521936875975974095538790244, 6.21814119013282252674914921853, 7.61862597488513330482060306993, 8.545799060836225515015131450227, 9.676041376288810321904334617728, 10.25088510867879273784801913454, 11.16182230154842535651975180164, 11.57258333834632449326666202455