| L(s) = 1 | + (1.35 + 0.415i)2-s + (0.0980 − 0.995i)3-s + (1.65 + 1.12i)4-s + (−0.677 + 0.361i)5-s + (0.545 − 1.30i)6-s + (0.993 + 4.99i)7-s + (1.77 + 2.20i)8-s + (−0.980 − 0.195i)9-s + (−1.06 + 0.208i)10-s + (0.499 − 0.609i)11-s + (1.27 − 1.53i)12-s + (−0.998 + 1.86i)13-s + (−0.730 + 7.16i)14-s + (0.293 + 0.709i)15-s + (1.47 + 3.71i)16-s + (2.93 − 7.09i)17-s + ⋯ |
| L(s) = 1 | + (0.955 + 0.293i)2-s + (0.0565 − 0.574i)3-s + (0.827 + 0.561i)4-s + (−0.302 + 0.161i)5-s + (0.222 − 0.532i)6-s + (0.375 + 1.88i)7-s + (0.626 + 0.779i)8-s + (−0.326 − 0.0650i)9-s + (−0.337 + 0.0658i)10-s + (0.150 − 0.183i)11-s + (0.369 − 0.443i)12-s + (−0.276 + 0.518i)13-s + (−0.195 + 1.91i)14-s + (0.0758 + 0.183i)15-s + (0.369 + 0.929i)16-s + (0.712 − 1.72i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.757 - 0.653i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.757 - 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.23608 + 0.831463i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.23608 + 0.831463i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.35 - 0.415i)T \) |
| 3 | \( 1 + (-0.0980 + 0.995i)T \) |
| good | 5 | \( 1 + (0.677 - 0.361i)T + (2.77 - 4.15i)T^{2} \) |
| 7 | \( 1 + (-0.993 - 4.99i)T + (-6.46 + 2.67i)T^{2} \) |
| 11 | \( 1 + (-0.499 + 0.609i)T + (-2.14 - 10.7i)T^{2} \) |
| 13 | \( 1 + (0.998 - 1.86i)T + (-7.22 - 10.8i)T^{2} \) |
| 17 | \( 1 + (-2.93 + 7.09i)T + (-12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (2.01 + 6.65i)T + (-15.7 + 10.5i)T^{2} \) |
| 23 | \( 1 + (-4.69 + 3.13i)T + (8.80 - 21.2i)T^{2} \) |
| 29 | \( 1 + (2.70 - 2.21i)T + (5.65 - 28.4i)T^{2} \) |
| 31 | \( 1 + (-3.85 + 3.85i)T - 31iT^{2} \) |
| 37 | \( 1 + (10.9 + 3.33i)T + (30.7 + 20.5i)T^{2} \) |
| 41 | \( 1 + (-4.07 - 6.09i)T + (-15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (0.587 + 5.96i)T + (-42.1 + 8.38i)T^{2} \) |
| 47 | \( 1 + (-1.54 - 0.637i)T + (33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (4.60 + 3.77i)T + (10.3 + 51.9i)T^{2} \) |
| 59 | \( 1 + (-1.77 - 3.32i)T + (-32.7 + 49.0i)T^{2} \) |
| 61 | \( 1 + (-6.20 - 0.611i)T + (59.8 + 11.9i)T^{2} \) |
| 67 | \( 1 + (-5.97 - 0.588i)T + (65.7 + 13.0i)T^{2} \) |
| 71 | \( 1 + (7.18 - 1.42i)T + (65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (-1.07 + 5.40i)T + (-67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (-1.18 + 0.491i)T + (55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (-2.28 + 0.691i)T + (69.0 - 46.1i)T^{2} \) |
| 89 | \( 1 + (-1.08 - 0.727i)T + (34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (6.78 - 6.78i)T - 97iT^{2} \) |
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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
−11.63695889524461338510670776116, −11.19502052082870127512237534911, −9.279790517920921450426360693475, −8.595928594951787210801025994212, −7.42117387494984834972841304491, −6.67715794438042268256570343507, −5.50425802065943408027919390746, −4.85476562360450196409398964225, −3.05641450910516386630194898957, −2.25198061420351334085978634906,
1.44437072968893352225169871252, 3.58802873419477859616404321932, 4.00166283787046300993265776800, 5.08264522553311629695002959183, 6.27532666827894260223498777034, 7.46152389536322464540547085337, 8.206685944473844479105752438604, 10.10443412815492782034187847758, 10.30161487456872771729676632527, 11.12980223948201493207046197498
