L(s) = 1 | + (−0.642 + 0.766i)2-s + (−0.556 − 1.64i)3-s + (−0.173 − 0.984i)4-s + (−0.428 + 0.359i)5-s + (1.61 + 0.628i)6-s + (−2.62 + 0.363i)7-s + (0.866 + 0.500i)8-s + (−2.38 + 1.82i)9-s − 0.558i·10-s + (0.831 − 0.991i)11-s + (−1.51 + 0.832i)12-s + (−2.35 + 6.46i)13-s + (1.40 − 2.24i)14-s + (0.827 + 0.502i)15-s + (−0.939 + 0.342i)16-s + 4.92·17-s + ⋯ |
L(s) = 1 | + (−0.454 + 0.541i)2-s + (−0.321 − 0.947i)3-s + (−0.0868 − 0.492i)4-s + (−0.191 + 0.160i)5-s + (0.658 + 0.256i)6-s + (−0.990 + 0.137i)7-s + (0.306 + 0.176i)8-s + (−0.793 + 0.608i)9-s − 0.176i·10-s + (0.250 − 0.298i)11-s + (−0.438 + 0.240i)12-s + (−0.652 + 1.79i)13-s + (0.375 − 0.598i)14-s + (0.213 + 0.129i)15-s + (−0.234 + 0.0855i)16-s + 1.19·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.469 - 0.882i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.469 - 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.219621 + 0.365469i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.219621 + 0.365469i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.642 - 0.766i)T \) |
| 3 | \( 1 + (0.556 + 1.64i)T \) |
| 7 | \( 1 + (2.62 - 0.363i)T \) |
good | 5 | \( 1 + (0.428 - 0.359i)T + (0.868 - 4.92i)T^{2} \) |
| 11 | \( 1 + (-0.831 + 0.991i)T + (-1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (2.35 - 6.46i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 - 4.92T + 17T^{2} \) |
| 19 | \( 1 - 0.427iT - 19T^{2} \) |
| 23 | \( 1 + (0.893 - 2.45i)T + (-17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.812 - 2.23i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (6.49 - 1.14i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (1.10 - 1.91i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (10.7 + 3.91i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (0.0502 - 0.284i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (0.476 - 2.70i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (-4.97 - 2.87i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (9.13 + 3.32i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (13.6 + 2.40i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-4.81 + 4.03i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-5.81 + 3.35i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (9.22 - 5.32i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.06 - 3.41i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-11.1 + 4.05i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 - 12.9T + 89T^{2} \) |
| 97 | \( 1 + (1.49 + 0.263i)T + (91.1 + 33.1i)T^{2} \) |
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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
−11.83295998222466308142536236642, −10.76663127803168502485005870183, −9.580119974272731480971611625798, −8.924450999561164531686951568064, −7.64689116917611339819636373062, −6.98228089771891348783819045285, −6.23500249560741626185671735521, −5.19952070202788687235252954812, −3.42881399576797128640463968722, −1.70419950250179135467184064868,
0.33978638987246674641680345085, 2.89886528799218718557226024459, 3.74841809362065890455426629217, 5.05226565414553179776350918954, 6.11592057068775016299837863659, 7.50716583454625902738500317541, 8.502332795101011358100841856075, 9.610147709031383155744745979716, 10.13611514330015952489050883604, 10.71242200683077321448052195769