| L(s) = 1 | + (−0.909 + 1.08i)2-s + (−0.347 − 1.96i)4-s + (2.44 + 1.41i)8-s + (−2.18 − 6.01i)11-s + (−3.75 + 1.36i)16-s + (−7.09 + 4.09i)17-s + (−0.511 + 0.885i)19-s + (8.50 + 3.09i)22-s + (−3.83 − 3.21i)25-s + (1.93 − 5.31i)32-s + (2.01 − 11.4i)34-s + (−0.494 − 1.35i)38-s + (−8.15 − 9.71i)41-s + (2.07 − 0.754i)43-s + (−11.0 + 6.39i)44-s + ⋯ |
| L(s) = 1 | + (−0.642 + 0.766i)2-s + (−0.173 − 0.984i)4-s + (0.866 + 0.500i)8-s + (−0.659 − 1.81i)11-s + (−0.939 + 0.342i)16-s + (−1.72 + 0.993i)17-s + (−0.117 + 0.203i)19-s + (1.81 + 0.659i)22-s + (−0.766 − 0.642i)25-s + (0.342 − 0.939i)32-s + (0.344 − 1.95i)34-s + (−0.0802 − 0.220i)38-s + (−1.27 − 1.51i)41-s + (0.316 − 0.115i)43-s + (−1.67 + 0.964i)44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.239 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.239 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.236182 - 0.301551i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.236182 - 0.301551i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.909 - 1.08i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (2.18 + 6.01i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (7.09 - 4.09i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.511 - 0.885i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (8.15 + 9.71i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-2.07 + 0.754i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + (-3.84 + 10.5i)T + (-45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (4.53 - 3.80i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.68 + 8.12i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-10.9 + 13.0i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (-11.0 - 6.38i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (14.0 - 5.12i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.41310054671751245533065737482, −9.149518656194040235834303004912, −8.476486255789054753359208064622, −7.921505668963604934543533995491, −6.62420093226408078640921985034, −6.03568116431922493783776113266, −5.06164786763884190959430292744, −3.75634681392634379741197381283, −2.09607416680917328768306197622, −0.24086844756557665918046530848,
1.84810314236789930441772259335, 2.76845288684522660434529357865, 4.27324542921197969843080313821, 4.96148285216211518015374718005, 6.74252013982445554764201055913, 7.38206827232492566720889666348, 8.312093451907088753084505742762, 9.386866537648557039073748770470, 9.802968396037808317085651695363, 10.79161007611481815312078912260
