| L(s) = 1 | + (1.97 + 1.49i)2-s + (0.468 + 0.883i)3-s + (1.10 + 3.99i)4-s + (−2.35 − 2.22i)5-s + (−0.400 + 2.44i)6-s + (0.391 − 0.460i)7-s + (−1.96 + 4.93i)8-s + (−0.561 + 0.827i)9-s + (−1.29 − 7.92i)10-s + (3.65 + 2.19i)11-s + (−3.00 + 2.84i)12-s + (−1.77 − 2.62i)13-s + (1.46 − 0.322i)14-s + (0.866 − 3.11i)15-s + (−4.18 + 2.51i)16-s + (−3.32 − 3.91i)17-s + ⋯ |
| L(s) = 1 | + (1.39 + 1.06i)2-s + (0.270 + 0.510i)3-s + (0.554 + 1.99i)4-s + (−1.05 − 0.995i)5-s + (−0.163 + 0.998i)6-s + (0.147 − 0.174i)7-s + (−0.695 + 1.74i)8-s + (−0.187 + 0.275i)9-s + (−0.410 − 2.50i)10-s + (1.10 + 0.662i)11-s + (−0.868 + 0.822i)12-s + (−0.493 − 0.727i)13-s + (0.391 − 0.0861i)14-s + (0.223 − 0.805i)15-s + (−1.04 + 0.629i)16-s + (−0.806 − 0.949i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00184 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00184 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.53507 + 1.53791i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.53507 + 1.53791i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.468 - 0.883i)T \) |
| 59 | \( 1 + (5.16 - 5.68i)T \) |
| good | 2 | \( 1 + (-1.97 - 1.49i)T + (0.535 + 1.92i)T^{2} \) |
| 5 | \( 1 + (2.35 + 2.22i)T + (0.270 + 4.99i)T^{2} \) |
| 7 | \( 1 + (-0.391 + 0.460i)T + (-1.13 - 6.90i)T^{2} \) |
| 11 | \( 1 + (-3.65 - 2.19i)T + (5.15 + 9.71i)T^{2} \) |
| 13 | \( 1 + (1.77 + 2.62i)T + (-4.81 + 12.0i)T^{2} \) |
| 17 | \( 1 + (3.32 + 3.91i)T + (-2.75 + 16.7i)T^{2} \) |
| 19 | \( 1 + (-3.35 + 1.54i)T + (12.3 - 14.4i)T^{2} \) |
| 23 | \( 1 + (6.29 + 2.11i)T + (18.3 + 13.9i)T^{2} \) |
| 29 | \( 1 + (3.47 - 2.64i)T + (7.75 - 27.9i)T^{2} \) |
| 31 | \( 1 + (-7.92 - 3.66i)T + (20.0 + 23.6i)T^{2} \) |
| 37 | \( 1 + (-1.67 - 4.21i)T + (-26.8 + 25.4i)T^{2} \) |
| 41 | \( 1 + (4.76 - 1.60i)T + (32.6 - 24.8i)T^{2} \) |
| 43 | \( 1 + (10.7 - 6.47i)T + (20.1 - 37.9i)T^{2} \) |
| 47 | \( 1 + (2.28 - 2.16i)T + (2.54 - 46.9i)T^{2} \) |
| 53 | \( 1 + (-2.21 + 13.4i)T + (-50.2 - 16.9i)T^{2} \) |
| 61 | \( 1 + (-0.0209 - 0.0159i)T + (16.3 + 58.7i)T^{2} \) |
| 67 | \( 1 + (-2.59 + 6.50i)T + (-48.6 - 46.0i)T^{2} \) |
| 71 | \( 1 + (-2.27 + 2.15i)T + (3.84 - 70.8i)T^{2} \) |
| 73 | \( 1 + (-2.20 + 0.484i)T + (66.2 - 30.6i)T^{2} \) |
| 79 | \( 1 + (-5.18 + 9.77i)T + (-44.3 - 65.3i)T^{2} \) |
| 83 | \( 1 + (3.33 - 0.362i)T + (81.0 - 17.8i)T^{2} \) |
| 89 | \( 1 + (-1.02 + 0.777i)T + (23.8 - 85.7i)T^{2} \) |
| 97 | \( 1 + (-4.15 - 0.914i)T + (88.0 + 40.7i)T^{2} \) |
| 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
−13.12450904007363196647901339634, −12.08268875582383094313600760804, −11.64998677519832990827261843496, −9.745112536739603335653726779043, −8.467220779211920274039269294991, −7.65707990715888908209501222508, −6.57752464827409237619187513831, −4.89133463615385768321695856571, −4.56824650695907882403006751894, −3.35591732327789639935594291656,
2.03995463918302573471805878810, 3.48396792845036974691688120277, 4.15290412899430864222804884095, 5.95842213277968231216252152005, 6.89980468377647516302021010303, 8.261592463499938379152156221343, 9.846118555372262477066909343999, 11.07183545700351324484532846958, 11.77930545490005340271836476027, 12.08843550284914358515974136942
