| L(s) = 1 | + (0.766 + 0.642i)2-s + (−1.36 + 1.07i)3-s + (0.173 + 0.984i)4-s + (0.696 + 0.253i)5-s + (−1.73 − 0.0539i)6-s + (0.717 − 4.07i)7-s + (−0.500 + 0.866i)8-s + (0.703 − 2.91i)9-s + (0.370 + 0.641i)10-s + (−4.27 + 1.55i)11-s + (−1.29 − 1.15i)12-s + (0.662 − 0.556i)13-s + (3.16 − 2.65i)14-s + (−1.21 + 0.401i)15-s + (−0.939 + 0.342i)16-s + (2.17 + 3.77i)17-s + ⋯ |
| L(s) = 1 | + (0.541 + 0.454i)2-s + (−0.785 + 0.618i)3-s + (0.0868 + 0.492i)4-s + (0.311 + 0.113i)5-s + (−0.706 − 0.0220i)6-s + (0.271 − 1.53i)7-s + (−0.176 + 0.306i)8-s + (0.234 − 0.972i)9-s + (0.117 + 0.202i)10-s + (−1.28 + 0.468i)11-s + (−0.372 − 0.333i)12-s + (0.183 − 0.154i)13-s + (0.846 − 0.709i)14-s + (−0.314 + 0.103i)15-s + (−0.234 + 0.0855i)16-s + (0.528 + 0.915i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.625 - 0.780i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.625 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.822422 + 0.395042i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.822422 + 0.395042i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.36 - 1.07i)T \) |
| 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
−15.60792211901933189114972986063, −14.49679005561854283864826627467, −13.38226561288601886899943686693, −12.27438631536301560065151817138, −10.65490836344145337633912221483, −10.20068395613446654696349418484, −8.002830441724452405031123696588, −6.65158643732470709345400657438, −5.22042601629672777934718492782, −3.94215172792586259298413553251,
2.39363991817582668294811212740, 5.24154657136006027338953748870, 5.88484242553135047079676747998, 7.80460859551780265837531810505, 9.481003735007022639138362787149, 11.06122960206069902803615442099, 11.83110003741093156904671035201, 12.88134730037421261247844071945, 13.71402322914293604695564151867, 15.26553877769395835152167961989
