| L(s) = 1 | + (0.00187 + 0.000630i)2-s + (1.71 − 0.275i)3-s + (−1.59 − 1.21i)4-s + (−2.20 − 0.878i)5-s + (0.00337 + 0.000563i)6-s + (2.80 − 1.29i)7-s + (−0.00443 − 0.00654i)8-s + (2.84 − 0.941i)9-s + (−0.00357 − 0.00303i)10-s + (1.08 − 3.91i)11-s + (−3.05 − 1.63i)12-s + (1.98 + 1.05i)13-s + (0.00607 − 0.000660i)14-s + (−4.01 − 0.895i)15-s + (1.07 + 3.85i)16-s + (−1.79 + 3.88i)17-s + ⋯ |
| L(s) = 1 | + (0.00132 + 0.000446i)2-s + (0.987 − 0.159i)3-s + (−0.796 − 0.605i)4-s + (−0.986 − 0.393i)5-s + (0.00137 + 0.000229i)6-s + (1.06 − 0.490i)7-s + (−0.00156 − 0.00231i)8-s + (0.949 − 0.313i)9-s + (−0.00113 − 0.000960i)10-s + (0.328 − 1.18i)11-s + (−0.882 − 0.470i)12-s + (0.551 + 0.292i)13-s + (0.00162 − 0.000176i)14-s + (−1.03 − 0.231i)15-s + (0.267 + 0.963i)16-s + (−0.436 + 0.943i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.510 + 0.860i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.510 + 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.10751 - 0.630806i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.10751 - 0.630806i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.71 + 0.275i)T \) |
| 59 | \( 1 + (-5.98 + 4.81i)T \) |
| good | 2 | \( 1 + (-0.00187 - 0.000630i)T + (1.59 + 1.21i)T^{2} \) |
| 5 | \( 1 + (2.20 + 0.878i)T + (3.62 + 3.43i)T^{2} \) |
| 7 | \( 1 + (-2.80 + 1.29i)T + (4.53 - 5.33i)T^{2} \) |
| 11 | \( 1 + (-1.08 + 3.91i)T + (-9.42 - 5.67i)T^{2} \) |
| 13 | \( 1 + (-1.98 - 1.05i)T + (7.29 + 10.7i)T^{2} \) |
| 17 | \( 1 + (1.79 - 3.88i)T + (-11.0 - 12.9i)T^{2} \) |
| 19 | \( 1 + (4.97 - 1.09i)T + (17.2 - 7.97i)T^{2} \) |
| 23 | \( 1 + (0.913 - 5.57i)T + (-21.7 - 7.34i)T^{2} \) |
| 29 | \( 1 + (-1.70 - 5.05i)T + (-23.0 + 17.5i)T^{2} \) |
| 31 | \( 1 + (-1.17 + 5.33i)T + (-28.1 - 13.0i)T^{2} \) |
| 37 | \( 1 + (-4.91 - 3.33i)T + (13.6 + 34.3i)T^{2} \) |
| 41 | \( 1 + (-8.65 + 1.41i)T + (38.8 - 13.0i)T^{2} \) |
| 43 | \( 1 + (2.54 - 0.706i)T + (36.8 - 22.1i)T^{2} \) |
| 47 | \( 1 + (-1.16 - 2.91i)T + (-34.1 + 32.3i)T^{2} \) |
| 53 | \( 1 + (5.66 - 4.81i)T + (8.57 - 52.3i)T^{2} \) |
| 61 | \( 1 + (-0.720 + 2.13i)T + (-48.5 - 36.9i)T^{2} \) |
| 67 | \( 1 + (12.7 - 8.61i)T + (24.7 - 62.2i)T^{2} \) |
| 71 | \( 1 + (-4.65 + 1.85i)T + (51.5 - 48.8i)T^{2} \) |
| 73 | \( 1 + (0.221 + 2.03i)T + (-71.2 + 15.6i)T^{2} \) |
| 79 | \( 1 + (-7.06 + 4.24i)T + (37.0 - 69.7i)T^{2} \) |
| 83 | \( 1 + (0.618 + 11.4i)T + (-82.5 + 8.97i)T^{2} \) |
| 89 | \( 1 + (15.9 - 5.38i)T + (70.8 - 53.8i)T^{2} \) |
| 97 | \( 1 + (-0.453 + 4.16i)T + (-94.7 - 20.8i)T^{2} \) |
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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
−12.82028864778636404783650578682, −11.42031438832824627090813852389, −10.60513802033791315579205136511, −9.177780945618521592248491895768, −8.380102101924748915525598676444, −7.88644500448987989306806385837, −6.15441428296348603888870106423, −4.42372287837967237180813061963, −3.83847097625380757093712586391, −1.34444127741198793025450195988,
2.53300377760701813890844242902, 4.09216935495267528101267605074, 4.67765009065523252079099610587, 7.03458627418589205193457659745, 8.021793648125463449498255340909, 8.563631821200683048152949145413, 9.590474801333870018008646546830, 10.93188953456317813640779406424, 12.03014230318533170151089992330, 12.83323817872757253607060130282
