L(s) = 1 | + i·5-s + (3.75 − 2.16i)7-s + (−1.5 − 0.866i)11-s + (−3.11 + 1.81i)13-s + (−3.75 − 6.49i)17-s + (4.65 − 2.68i)19-s + (0.580 − 1.00i)23-s − 25-s + (−1.01 + 1.75i)29-s − 7.86i·31-s + (2.16 + 3.75i)35-s + (−8.25 − 4.76i)37-s + (−6.69 − 3.86i)41-s + (2.09 + 3.62i)43-s + 3.46i·47-s + ⋯ |
L(s) = 1 | + 0.447i·5-s + (1.41 − 0.818i)7-s + (−0.452 − 0.261i)11-s + (−0.863 + 0.504i)13-s + (−0.909 − 1.57i)17-s + (1.06 − 0.616i)19-s + (0.121 − 0.209i)23-s − 0.200·25-s + (−0.187 + 0.325i)29-s − 1.41i·31-s + (0.366 + 0.634i)35-s + (−1.35 − 0.783i)37-s + (−1.04 − 0.603i)41-s + (0.319 + 0.552i)43-s + 0.505i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0183 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0183 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.545112233\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.545112233\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 + (3.11 - 1.81i)T \) |
good | 7 | \( 1 + (-3.75 + 2.16i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.5 + 0.866i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (3.75 + 6.49i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.65 + 2.68i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.580 + 1.00i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.01 - 1.75i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 7.86iT - 31T^{2} \) |
| 37 | \( 1 + (8.25 + 4.76i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (6.69 + 3.86i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.09 - 3.62i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 3.46iT - 47T^{2} \) |
| 53 | \( 1 - 12.6T + 53T^{2} \) |
| 59 | \( 1 + (5.49 - 3.17i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.85 + 3.20i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4.55 + 2.63i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-10.8 + 6.25i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 5.23iT - 73T^{2} \) |
| 79 | \( 1 + 8.16T + 79T^{2} \) |
| 83 | \( 1 + 0.456iT - 83T^{2} \) |
| 89 | \( 1 + (-11.4 - 6.63i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (2.43 - 1.40i)T + (48.5 - 84.0i)T^{2} \) |
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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
−8.859423102731440457911656858977, −7.76864298218381497566519480135, −7.34222901139307698507276665471, −6.77817712810029895081391452276, −5.36848917740848284651872950764, −4.88225686992551987572719142252, −4.06747792630849771328685999325, −2.83049186614492714499743631491, −1.96428241252385592707180600001, −0.51371757724103574672679695746,
1.47776771459947422870309231032, 2.19790727254692922148629821142, 3.45032400570871732344200245496, 4.64284015304572472447476489750, 5.18225639567922743596296999538, 5.79065427587769642796769635508, 6.99789645707214973702020589549, 7.82001338035085302520456196218, 8.463647934406847022787943140513, 8.877136064611095856661589467395