| L(s) = 1 | + (0.457 − 0.264i)2-s + (−1.86 + 3.22i)4-s + (−1.38 − 2.39i)7-s + 4.08i·8-s + (7.99 − 4.61i)11-s + (−6.79 + 11.7i)13-s + (−1.26 − 0.731i)14-s + (−6.36 − 11.0i)16-s + 12.2i·17-s − 20.2·19-s + (2.43 − 4.22i)22-s + (−2.05 − 1.18i)23-s + 7.18i·26-s + 10.2·28-s + (30.2 − 17.4i)29-s + ⋯ |
| L(s) = 1 | + (0.228 − 0.132i)2-s + (−0.465 + 0.805i)4-s + (−0.197 − 0.342i)7-s + 0.510i·8-s + (0.726 − 0.419i)11-s + (−0.522 + 0.905i)13-s + (−0.0904 − 0.0522i)14-s + (−0.397 − 0.688i)16-s + 0.718i·17-s − 1.06·19-s + (0.110 − 0.192i)22-s + (−0.0892 − 0.0515i)23-s + 0.276i·26-s + 0.367·28-s + (1.04 − 0.601i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00852i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 - 0.00852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3460774049\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3460774049\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + (-0.457 + 0.264i)T + (2 - 3.46i)T^{2} \) |
| 7 | \( 1 + (1.38 + 2.39i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-7.99 + 4.61i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (6.79 - 11.7i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 - 12.2iT - 289T^{2} \) |
| 19 | \( 1 + 20.2T + 361T^{2} \) |
| 23 | \( 1 + (2.05 + 1.18i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-30.2 + 17.4i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (14.7 - 25.5i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + 64.3T + 1.36e3T^{2} \) |
| 41 | \( 1 + (34.5 + 19.9i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (33.7 + 58.5i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (80.8 - 46.6i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 - 9.82iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (50.6 + 29.2i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-7.75 - 13.4i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-7.78 + 13.4i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 53.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 23.6T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-17.2 - 29.9i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (65.2 - 37.6i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 - 29.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (31.1 + 54.0i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−10.75477465518636463443722433453, −9.857668171968093388147146067228, −8.760345849378072802284287415407, −8.402690216500962605022195301525, −7.10728251018830891647240951566, −6.45588348959467466110353162174, −5.04089682468210736878829839143, −4.11613755794203260715131891822, −3.36901458168710350375858477466, −1.90933799939115721989664211274,
0.11224429400179295998414035093, 1.68983435835903265665761853094, 3.19370028467557535174468491081, 4.51244967766986156311414357959, 5.21344242487303332622850571942, 6.25858557128659532460998169184, 6.97006849944573660692113008289, 8.256911217951841876762208409698, 9.109518722351270878201401847367, 9.898188317464680805056081922317
