L(s) = 1 | + (1.38 + 1.90i)2-s + (0.951 − 0.309i)3-s + (−1.09 + 3.37i)4-s + (1.90 + 1.38i)6-s + (0.184 + 0.0598i)7-s + (−3.47 + 1.12i)8-s + (0.809 − 0.587i)9-s + (−1.96 + 2.67i)11-s + 3.54i·12-s + (0.572 + 0.787i)13-s + (0.140 + 0.433i)14-s + (−1.21 − 0.880i)16-s + (−1.57 + 2.16i)17-s + (2.24 + 0.727i)18-s + (1.71 + 5.27i)19-s + ⋯ |
L(s) = 1 | + (0.979 + 1.34i)2-s + (0.549 − 0.178i)3-s + (−0.548 + 1.68i)4-s + (0.778 + 0.565i)6-s + (0.0695 + 0.0226i)7-s + (−1.22 + 0.398i)8-s + (0.269 − 0.195i)9-s + (−0.591 + 0.806i)11-s + 1.02i·12-s + (0.158 + 0.218i)13-s + (0.0376 + 0.115i)14-s + (−0.303 − 0.220i)16-s + (−0.381 + 0.525i)17-s + (0.528 + 0.171i)18-s + (0.393 + 1.21i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.692 - 0.721i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.692 - 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.16128 + 2.72270i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.16128 + 2.72270i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.951 + 0.309i)T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + (1.96 - 2.67i)T \) |
good | 2 | \( 1 + (-1.38 - 1.90i)T + (-0.618 + 1.90i)T^{2} \) |
| 7 | \( 1 + (-0.184 - 0.0598i)T + (5.66 + 4.11i)T^{2} \) |
| 13 | \( 1 + (-0.572 - 0.787i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (1.57 - 2.16i)T + (-5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-1.71 - 5.27i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 - 4.80iT - 23T^{2} \) |
| 29 | \( 1 + (-3.12 + 9.62i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-2.02 + 1.47i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-5.43 - 1.76i)T + (29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (2.55 + 7.87i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 5.11iT - 43T^{2} \) |
| 47 | \( 1 + (-10.3 + 3.35i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (5.45 + 7.51i)T + (-16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-3.46 + 10.6i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (0.975 + 0.708i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 - 3.25iT - 67T^{2} \) |
| 71 | \( 1 + (4.84 + 3.52i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (3.14 + 1.02i)T + (59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-8.21 + 5.96i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (4.88 - 6.72i)T + (-25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + 7.34T + 89T^{2} \) |
| 97 | \( 1 + (-9.31 - 12.8i)T + (-29.9 + 92.2i)T^{2} \) |
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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
−10.33970416906626408936497840043, −9.548881156121468651336381861069, −8.311459669286819986012221879333, −7.86321564028996318682544610111, −7.06325126430318873060950019251, −6.17924599368187610766411618545, −5.35366537243136498291601504210, −4.32320276398499548446586418560, −3.56576707437200235818769455123, −2.06948182664867249031606663320,
1.09136142090805271333771906692, 2.77321713814794721611007288589, 2.98033445554917942538480758348, 4.39588912278513227244981352667, 4.96402165255606995726419634065, 6.08682364308539708938666376053, 7.35156358742935664945870903634, 8.496851896599185107683095137547, 9.261573615812598868499007383422, 10.23317192765892593918393773762