L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.866 − 0.5i)3-s + (−0.499 + 0.866i)4-s + (−0.465 − 2.18i)5-s − 0.999i·6-s + (−3.89 − 2.24i)7-s − 0.999·8-s + (0.499 + 0.866i)9-s + (1.66 − 1.49i)10-s + 0.412·11-s + (0.866 − 0.499i)12-s + (−1.10 + 1.90i)13-s − 4.49i·14-s + (−0.690 + 2.12i)15-s + (−0.5 − 0.866i)16-s + (1.63 + 2.82i)17-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.499 − 0.288i)3-s + (−0.249 + 0.433i)4-s + (−0.208 − 0.978i)5-s − 0.408i·6-s + (−1.47 − 0.849i)7-s − 0.353·8-s + (0.166 + 0.288i)9-s + (0.525 − 0.473i)10-s + 0.124·11-s + (0.249 − 0.144i)12-s + (−0.305 + 0.529i)13-s − 1.20i·14-s + (−0.178 + 0.549i)15-s + (−0.125 − 0.216i)16-s + (0.395 + 0.685i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.531 - 0.847i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.531 - 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5869313394\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5869313394\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 + (0.866 + 0.5i)T \) |
| 5 | \( 1 + (0.465 + 2.18i)T \) |
| 37 | \( 1 + (5.42 + 2.74i)T \) |
good | 7 | \( 1 + (3.89 + 2.24i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 - 0.412T + 11T^{2} \) |
| 13 | \( 1 + (1.10 - 1.90i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.63 - 2.82i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.06 - 1.19i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 1.63T + 23T^{2} \) |
| 29 | \( 1 - 9.16iT - 29T^{2} \) |
| 31 | \( 1 - 6.51iT - 31T^{2} \) |
| 41 | \( 1 + (1.68 - 2.91i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 - 9.32T + 43T^{2} \) |
| 47 | \( 1 + 6.71iT - 47T^{2} \) |
| 53 | \( 1 + (10.0 - 5.82i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.76 + 2.74i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.46 - 3.73i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.71 + 1.56i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (1.01 - 1.75i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 6.57iT - 73T^{2} \) |
| 79 | \( 1 + (7.72 + 4.46i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (3.38 - 1.95i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (2.49 - 1.44i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 10.9T + 97T^{2} \) |
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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
−10.01627509404186968494119760126, −9.223816296040754920370082768571, −8.434405103674193087643232717957, −7.27191663445345447381059050194, −6.89270859520045895097402877525, −5.89974464833672277820669861571, −5.09930844864079064599248622283, −4.08942760821400147312905067687, −3.28254870567697993927185544844, −1.24813071472892844394455597706,
0.26608568670221778351655977221, 2.52520612797372886574608972934, 3.11970050524882287460559262053, 4.07663528903173792106056032931, 5.38701583705658048620665188986, 6.05832996623631490872801452220, 6.79663183129669124254443932926, 7.81020626064628188777367292765, 9.229003641606287831863476760793, 9.747193421488904320123076594968