| L(s) = 1 | + (−1.85 − 0.162i)2-s + (1.46 + 0.257i)4-s + (2.23 + 0.123i)5-s + (−2.56 − 1.79i)7-s + (0.928 + 0.248i)8-s + (−4.13 − 0.592i)10-s + (1.21 − 3.34i)11-s + (0.165 + 1.89i)13-s + (4.47 + 3.75i)14-s + (−4.47 − 1.62i)16-s + (−3.30 + 0.886i)17-s + (5.00 − 2.89i)19-s + (3.23 + 0.756i)20-s + (−2.80 + 6.02i)22-s + (−0.749 − 1.06i)23-s + ⋯ |
| L(s) = 1 | + (−1.31 − 0.115i)2-s + (0.731 + 0.128i)4-s + (0.998 + 0.0552i)5-s + (−0.969 − 0.678i)7-s + (0.328 + 0.0879i)8-s + (−1.30 − 0.187i)10-s + (0.367 − 1.00i)11-s + (0.0460 + 0.525i)13-s + (1.19 + 1.00i)14-s + (−1.11 − 0.407i)16-s + (−0.802 + 0.215i)17-s + (1.14 − 0.663i)19-s + (0.723 + 0.169i)20-s + (−0.598 + 1.28i)22-s + (−0.156 − 0.223i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.253 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.253 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.538601 - 0.415492i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.538601 - 0.415492i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-2.23 - 0.123i)T \) |
| good | 2 | \( 1 + (1.85 + 0.162i)T + (1.96 + 0.347i)T^{2} \) |
| 7 | \( 1 + (2.56 + 1.79i)T + (2.39 + 6.57i)T^{2} \) |
| 11 | \( 1 + (-1.21 + 3.34i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-0.165 - 1.89i)T + (-12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (3.30 - 0.886i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-5.00 + 2.89i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.749 + 1.06i)T + (-7.86 + 21.6i)T^{2} \) |
| 29 | \( 1 + (0.0144 - 0.0121i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.260 + 1.47i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (2.61 + 9.77i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-6.39 + 7.61i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (3.23 + 6.93i)T + (-27.6 + 32.9i)T^{2} \) |
| 47 | \( 1 + (-6.31 + 9.02i)T + (-16.0 - 44.1i)T^{2} \) |
| 53 | \( 1 + (3.54 - 3.54i)T - 53iT^{2} \) |
| 59 | \( 1 + (-5.27 + 1.91i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.47 - 8.37i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-0.222 + 0.0194i)T + (65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (-0.428 - 0.247i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.07 - 7.75i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-1.13 - 1.35i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (0.643 - 7.35i)T + (-81.7 - 14.4i)T^{2} \) |
| 89 | \( 1 + (4.43 + 7.67i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.93 + 3.23i)T + (62.3 - 74.3i)T^{2} \) |
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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
−10.68658278041718263332090377601, −10.07576041039615085667205552434, −9.111025776406099894793310699477, −8.845230760449240072863448280474, −7.30957208074679464967824696100, −6.66493146078944401861354353472, −5.51140792265407568693116947557, −3.87385235810825155847511428489, −2.31937006180602839406610535859, −0.73095238815773265513842224489,
1.50440156933454808036514484149, 2.88073807420662788219699017632, 4.75296507831197467705657548786, 6.05881741237951936683019132637, 6.84262938741713398108271726620, 7.928896006284284760250674738995, 8.990582664159398782813853566393, 9.754530635723532654897743448142, 9.886774501167027895231197867097, 11.13486092405790604511151753640
