| L(s) = 1 | + (−1.29 − 0.560i)2-s + (0.335 + 1.69i)3-s + (1.37 + 1.45i)4-s + (−4.04 + 0.353i)5-s + (0.517 − 2.39i)6-s + (−0.174 − 0.0633i)7-s + (−0.963 − 2.65i)8-s + (−2.77 + 1.13i)9-s + (5.45 + 1.80i)10-s + (3.97 + 0.348i)11-s + (−2.01 + 2.81i)12-s + (−0.940 − 1.34i)13-s + (0.190 + 0.179i)14-s + (−1.95 − 6.75i)15-s + (−0.239 + 3.99i)16-s + (−6.25 − 3.61i)17-s + ⋯ |
| L(s) = 1 | + (−0.918 − 0.396i)2-s + (0.193 + 0.981i)3-s + (0.685 + 0.727i)4-s + (−1.80 + 0.158i)5-s + (0.211 − 0.977i)6-s + (−0.0657 − 0.0239i)7-s + (−0.340 − 0.940i)8-s + (−0.925 + 0.379i)9-s + (1.72 + 0.571i)10-s + (1.19 + 0.104i)11-s + (−0.581 + 0.813i)12-s + (−0.260 − 0.372i)13-s + (0.0509 + 0.0480i)14-s + (−0.505 − 1.74i)15-s + (−0.0598 + 0.998i)16-s + (−1.51 − 0.875i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.486 + 0.873i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.486 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0801702 - 0.136485i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0801702 - 0.136485i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.29 + 0.560i)T \) |
| 3 | \( 1 + (-0.335 - 1.69i)T \) |
| good | 5 | \( 1 + (4.04 - 0.353i)T + (4.92 - 0.868i)T^{2} \) |
| 7 | \( 1 + (0.174 + 0.0633i)T + (5.36 + 4.49i)T^{2} \) |
| 11 | \( 1 + (-3.97 - 0.348i)T + (10.8 + 1.91i)T^{2} \) |
| 13 | \( 1 + (0.940 + 1.34i)T + (-4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (6.25 + 3.61i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.550 - 2.05i)T + (-16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (0.510 + 1.40i)T + (-17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-3.64 + 5.20i)T + (-9.91 - 27.2i)T^{2} \) |
| 31 | \( 1 + (3.40 + 9.35i)T + (-23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (-2.30 + 0.617i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (0.875 - 4.96i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-0.345 + 3.94i)T + (-42.3 - 7.46i)T^{2} \) |
| 47 | \( 1 + (7.00 + 2.54i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (1.76 - 1.76i)T - 53iT^{2} \) |
| 59 | \( 1 + (-0.934 - 10.6i)T + (-58.1 + 10.2i)T^{2} \) |
| 61 | \( 1 + (2.60 - 1.21i)T + (39.2 - 46.7i)T^{2} \) |
| 67 | \( 1 + (1.64 - 1.15i)T + (22.9 - 62.9i)T^{2} \) |
| 71 | \( 1 + (5.42 + 3.12i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (8.58 - 4.95i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (8.88 - 1.56i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-5.36 - 3.75i)T + (28.3 + 77.9i)T^{2} \) |
| 89 | \( 1 + (-6.42 - 11.1i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-9.12 + 7.65i)T + (16.8 - 95.5i)T^{2} \) |
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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
−10.94457062196268849828126674142, −9.907987961497746306647887620091, −9.044899326807261720464505570127, −8.306799372599770033265829830298, −7.49970494261351556234724896031, −6.45756980057589967262361379347, −4.42023641034472001404250005317, −3.88185245869701089672733641736, −2.71142957746977573117288559201, −0.13473455849882089686311507326,
1.50607099626570441272197708750, 3.28012821271402693742789703376, 4.67430231983801808463540437596, 6.48656641862154283676372614021, 6.90397730894911817234646254639, 7.82401029053018512492863294160, 8.693547092787456136137787804109, 9.043436290391919694953372255646, 10.81826093097437382783430971618, 11.42711840624280189296699613265
