L(s) = 1 | + (−0.5 − 0.866i)2-s − 1.73i·3-s + (−0.499 + 0.866i)4-s + 2·5-s + (−1.49 + 0.866i)6-s + 0.999·8-s − 2.99·9-s + (−1 − 1.73i)10-s + (2 − 3.46i)11-s + (1.49 + 0.866i)12-s + (2 − 3.46i)13-s − 3.46i·15-s + (−0.5 − 0.866i)16-s + (1.5 − 2.59i)17-s + (1.49 + 2.59i)18-s + (−4 + 1.73i)19-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s − 0.999i·3-s + (−0.249 + 0.433i)4-s + 0.894·5-s + (−0.612 + 0.353i)6-s + 0.353·8-s − 0.999·9-s + (−0.316 − 0.547i)10-s + (0.603 − 1.04i)11-s + (0.433 + 0.249i)12-s + (0.554 − 0.960i)13-s − 0.894i·15-s + (−0.125 − 0.216i)16-s + (0.363 − 0.630i)17-s + (0.353 + 0.612i)18-s + (−0.917 + 0.397i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.564 + 0.825i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.564 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.562623 - 1.06691i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.562623 - 1.06691i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 + 1.73iT \) |
| 19 | \( 1 + (4 - 1.73i)T \) |
good | 5 | \( 1 - 2T + 5T^{2} \) |
| 7 | \( 1 + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2 + 3.46i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2 + 3.46i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (4 - 6.92i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 10T + 29T^{2} \) |
| 31 | \( 1 + (5 + 8.66i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - T + 41T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 2T + 47T^{2} \) |
| 53 | \( 1 + (-5 - 8.66i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 3T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + (-1.5 + 2.59i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-5 + 8.66i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (3 - 5.19i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-5 - 8.66i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (4.5 - 7.79i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-7 - 12.1i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.28814680400560968635643899506, −10.35134699556924724088916377135, −9.331186412028355193077405863752, −8.429142579393823071755941427891, −7.61108803843543669022223920469, −6.20347936479481287609309167473, −5.63954934615538305608528442083, −3.61634147661841155051653021787, −2.36231461627202374946129344364, −1.02671861081739667039206027722,
2.04371368732317582292535085613, 4.01446723617531949750335195564, 4.90714057368076423708069862005, 6.15463505771442292162601432201, 6.79015180245836128893274082336, 8.516505441039578049456935713377, 8.949796245085608830321915236558, 10.14886256059969758295479404161, 10.32677574692642200614154241085, 11.70280866107581820238458441196