| L(s) = 1 | + (−0.766 − 0.642i)2-s + (0.173 + 0.984i)4-s + (0.939 + 0.342i)5-s + (−0.220 + 1.24i)7-s + (0.500 − 0.866i)8-s + (−0.5 − 0.866i)10-s + (0.595 − 0.216i)11-s + (1.67 − 1.40i)13-s + (0.971 − 0.815i)14-s + (−0.939 + 0.342i)16-s + (−0.812 − 1.40i)17-s + (1.58 − 2.73i)19-s + (−0.173 + 0.984i)20-s + (−0.595 − 0.216i)22-s + (1.41 + 8.01i)23-s + ⋯ |
| L(s) = 1 | + (−0.541 − 0.454i)2-s + (0.0868 + 0.492i)4-s + (0.420 + 0.152i)5-s + (−0.0832 + 0.471i)7-s + (0.176 − 0.306i)8-s + (−0.158 − 0.273i)10-s + (0.179 − 0.0653i)11-s + (0.463 − 0.389i)13-s + (0.259 − 0.217i)14-s + (−0.234 + 0.0855i)16-s + (−0.197 − 0.341i)17-s + (0.362 − 0.628i)19-s + (−0.0388 + 0.220i)20-s + (−0.127 − 0.0462i)22-s + (0.294 + 1.67i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0104i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0104i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.29567 + 0.00677334i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.29567 + 0.00677334i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.766 + 0.642i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.939 - 0.342i)T \) |
| good | 7 | \( 1 + (0.220 - 1.24i)T + (-6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (-0.595 + 0.216i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-1.67 + 1.40i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (0.812 + 1.40i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.58 + 2.73i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.41 - 8.01i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-7.83 - 6.57i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (0.163 + 0.924i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-0.772 - 1.33i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.67 + 3.91i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-0.645 + 0.234i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (0.296 - 1.68i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 7.46T + 53T^{2} \) |
| 59 | \( 1 + (-9.43 - 3.43i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (0.981 - 5.56i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (4.57 - 3.83i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (3.18 + 5.52i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-6.81 + 11.8i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.15 - 2.64i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-12.6 - 10.6i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-7.25 + 12.5i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.93 + 2.52i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.25659093687696107749764140972, −9.289309093916516034644079667380, −8.883573294837748317342749438246, −7.77516072395187794483569880155, −6.90877175725303574427265642363, −5.89047027369217897873402213014, −4.91358969176097319878827552841, −3.48164524522599635102048900755, −2.58839807152494122808822850179, −1.20041334562274655522015358299,
0.964868503232666073957707688183, 2.39681516085878689903440107212, 3.97712280568979786584197742511, 4.95192041617987315367714310841, 6.22109525389849038426308708058, 6.61845788182869908579942618911, 7.82030052150901704074800342106, 8.515129134579078274906162921647, 9.368496739898677011519128302061, 10.18211938888837519459876419767
