L(s) = 1 | + (0.5 − 0.866i)5-s + (1 + 1.73i)7-s + (−1 − 1.73i)11-s + (−0.5 + 0.866i)13-s − 3·17-s − 2·19-s + (−3 + 5.19i)23-s + (2 + 3.46i)25-s + (0.5 + 0.866i)29-s + (−4 + 6.92i)31-s + 1.99·35-s + 37-s + (−1 + 1.73i)41-s + (5 + 8.66i)43-s + (−2 − 3.46i)47-s + ⋯ |
L(s) = 1 | + (0.223 − 0.387i)5-s + (0.377 + 0.654i)7-s + (−0.301 − 0.522i)11-s + (−0.138 + 0.240i)13-s − 0.727·17-s − 0.458·19-s + (−0.625 + 1.08i)23-s + (0.400 + 0.692i)25-s + (0.0928 + 0.160i)29-s + (−0.718 + 1.24i)31-s + 0.338·35-s + 0.164·37-s + (−0.156 + 0.270i)41-s + (0.762 + 1.32i)43-s + (−0.291 − 0.505i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.211379134\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.211379134\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.5 + 0.866i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-1 - 1.73i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.5 - 0.866i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (4 - 6.92i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - T + 37T^{2} \) |
| 41 | \( 1 + (1 - 1.73i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-5 - 8.66i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (2 + 3.46i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 10T + 53T^{2} \) |
| 59 | \( 1 + (2 - 3.46i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4.5 + 7.79i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (7 - 12.1i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 10T + 71T^{2} \) |
| 73 | \( 1 + 9T + 73T^{2} \) |
| 79 | \( 1 + (-5 - 8.66i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (6 + 10.3i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 11T + 89T^{2} \) |
| 97 | \( 1 + (-1 - 1.73i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.911255708781344729933663402176, −8.554807035496534996533316220208, −7.62108880815641104670994543659, −6.80271395109578091318660837622, −5.83874839468472582514060502831, −5.28442593520357941364323442290, −4.44554280208343950487755281782, −3.38574537237918447587381315370, −2.33832156473046897884886547834, −1.37365606148837335769490840013,
0.38776380686161852716118348098, 1.97459358961006231662105838409, 2.69544636619853081774063136119, 4.07703912151907414651163869830, 4.50693775624085513391284015302, 5.62301093562828267718676625827, 6.43018109204937428523431534766, 7.17320173651545764709100144144, 7.83675023880281529294798967000, 8.630154931725456396082664652481