| 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
−15.26553877769395835152167961989, −13.71402322914293604695564151867, −12.88134730037421261247844071945, −11.83110003741093156904671035201, −11.06122960206069902803615442099, −9.481003735007022639138362787149, −7.80460859551780265837531810505, −5.88484242553135047079676747998, −5.24154657136006027338953748870, −2.39363991817582668294811212740,
3.94215172792586259298413553251, 5.22042601629672777934718492782, 6.65158643732470709345400657438, 8.002830441724452405031123696588, 10.20068395613446654696349418484, 10.65490836344145337633912221483, 12.27438631536301560065151817138, 13.38226561288601886899943686693, 14.49679005561854283864826627467, 15.60792211901933189114972986063
