| L(s) = 1 | + (−0.766 + 0.642i)2-s + (0.173 − 0.984i)4-s + (−0.696 + 0.253i)5-s + (0.717 + 4.07i)7-s + (0.500 + 0.866i)8-s + (0.370 − 0.641i)10-s + (4.27 + 1.55i)11-s + (0.662 + 0.556i)13-s + (−3.16 − 2.65i)14-s + (−0.939 − 0.342i)16-s + (−2.17 + 3.77i)17-s + (−0.777 − 1.34i)19-s + (0.128 + 0.729i)20-s + (−4.27 + 1.55i)22-s + (0.608 − 3.45i)23-s + ⋯ |
| L(s) = 1 | + (−0.541 + 0.454i)2-s + (0.0868 − 0.492i)4-s + (−0.311 + 0.113i)5-s + (0.271 + 1.53i)7-s + (0.176 + 0.306i)8-s + (0.117 − 0.202i)10-s + (1.28 + 0.468i)11-s + (0.183 + 0.154i)13-s + (−0.846 − 0.709i)14-s + (−0.234 − 0.0855i)16-s + (−0.528 + 0.915i)17-s + (−0.178 − 0.309i)19-s + (0.0287 + 0.163i)20-s + (−0.910 + 0.331i)22-s + (0.126 − 0.719i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.226 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.226 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.681720 + 0.541313i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.681720 + 0.541313i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 + (0.696 - 0.253i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (-0.717 - 4.07i)T + (-6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (-4.27 - 1.55i)T + (8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-0.662 - 0.556i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (2.17 - 3.77i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.777 + 1.34i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.608 + 3.45i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-2.50 + 2.10i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-1.85 + 10.5i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-0.880 + 1.52i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.97 - 1.65i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-2.58 - 0.941i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (1.68 + 9.54i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + 4.00T + 53T^{2} \) |
| 59 | \( 1 + (-1.34 + 0.489i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (0.751 + 4.26i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-10.0 - 8.42i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (2.54 - 4.40i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.286 - 0.496i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.17 + 4.34i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (7.06 - 5.92i)T + (14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (6.19 + 10.7i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (5.40 + 1.96i)T + (74.3 + 62.3i)T^{2} \) |
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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
−12.97816158565260640814085562826, −11.86609588772670116901107650613, −11.24361914820460541422526751116, −9.737936474916575534883266805249, −8.909819929710662128719709364199, −8.107909568942988029641663453892, −6.67760951807602306568160694360, −5.80533554640683310625680682057, −4.24828266183325905609633950965, −2.12175435445863802592914378363,
1.13696009375944745073072174369, 3.47088859322739664258051467528, 4.52434656270271572057725237928, 6.56871457797831596926040186803, 7.51746538118268451556309985506, 8.609068582659702761404544614418, 9.683009635056238934114608850208, 10.74595356850196041809654398479, 11.44641653606729768339749291949, 12.44174220314901144385439980309
