L(s) = 1 | + (0.308 − 2.21i)5-s + (4.27 − 2.46i)7-s + (1.20 + 2.08i)11-s + (2.51 + 1.45i)13-s + 6.86i·17-s + 4.17·19-s + (2.90 + 1.67i)23-s + (−4.80 − 1.36i)25-s + (2.59 + 4.5i)29-s + (3.08 − 5.35i)31-s + (−4.14 − 10.2i)35-s + 7.84i·37-s + (2.93 − 5.08i)41-s + (−4.27 + 2.46i)43-s + (−10.3 + 5.95i)47-s + ⋯ |
L(s) = 1 | + (0.138 − 0.990i)5-s + (1.61 − 0.932i)7-s + (0.363 + 0.629i)11-s + (0.698 + 0.403i)13-s + 1.66i·17-s + 0.958·19-s + (0.606 + 0.350i)23-s + (−0.961 − 0.273i)25-s + (0.482 + 0.835i)29-s + (0.554 − 0.961i)31-s + (−0.700 − 1.72i)35-s + 1.28i·37-s + (0.458 − 0.794i)41-s + (−0.651 + 0.376i)43-s + (−1.50 + 0.868i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.883 + 0.468i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.883 + 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.341389876\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.341389876\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.308 + 2.21i)T \) |
good | 7 | \( 1 + (-4.27 + 2.46i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.20 - 2.08i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.51 - 1.45i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 6.86iT - 17T^{2} \) |
| 19 | \( 1 - 4.17T + 19T^{2} \) |
| 23 | \( 1 + (-2.90 - 1.67i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.59 - 4.5i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.08 + 5.35i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 7.84iT - 37T^{2} \) |
| 41 | \( 1 + (-2.93 + 5.08i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (4.27 - 2.46i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (10.3 - 5.95i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 8.54iT - 53T^{2} \) |
| 59 | \( 1 + (0.525 - 0.910i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4.58 + 7.94i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.50 + 2.02i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 + 2.02iT - 73T^{2} \) |
| 79 | \( 1 + (-3 - 5.19i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.49 + 2.59i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 3.09T + 89T^{2} \) |
| 97 | \( 1 + (-0.764 + 0.441i)T + (48.5 - 84.0i)T^{2} \) |
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
−9.278881822588788919218843421922, −8.292946592572258349932101292231, −8.047197575669937802455277186787, −7.02567416904860823506675966486, −6.02493337196784446309241527351, −4.93757089679416917011854679376, −4.49810809932909906376065160102, −3.60208271303249083282870283839, −1.64446217330760688797560269864, −1.30074748956038411834469220629,
1.22042631456308810225506503996, 2.52896332803372647864831788050, 3.22166577538490285742243658019, 4.61019315754022626492346038303, 5.39732928430481267185242724750, 6.12429020590088879866274821370, 7.15379774285037595464904300807, 7.86442840656696216529411914228, 8.659130019093059080724265263629, 9.321557578182991423631258327871