L(s) = 1 | + (−1.37 + 0.334i)2-s + (1.77 − 0.920i)4-s + 4.18·5-s + i·7-s + (−2.13 + 1.85i)8-s + (−5.74 + 1.40i)10-s − 1.86i·11-s + 3.07i·13-s + (−0.334 − 1.37i)14-s + (2.30 − 3.26i)16-s + 0.504i·17-s + 3.05·19-s + (7.42 − 3.84i)20-s + (0.625 + 2.56i)22-s − 5.97·23-s + ⋯ |
L(s) = 1 | + (−0.971 + 0.236i)2-s + (0.887 − 0.460i)4-s + 1.87·5-s + 0.377i·7-s + (−0.753 + 0.657i)8-s + (−1.81 + 0.443i)10-s − 0.563i·11-s + 0.853i·13-s + (−0.0895 − 0.367i)14-s + (0.576 − 0.817i)16-s + 0.122i·17-s + 0.701·19-s + (1.66 − 0.860i)20-s + (0.133 + 0.546i)22-s − 1.24·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.753 - 0.657i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.753 - 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.588189281\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.588189281\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.37 - 0.334i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 5 | \( 1 - 4.18T + 5T^{2} \) |
| 11 | \( 1 + 1.86iT - 11T^{2} \) |
| 13 | \( 1 - 3.07iT - 13T^{2} \) |
| 17 | \( 1 - 0.504iT - 17T^{2} \) |
| 19 | \( 1 - 3.05T + 19T^{2} \) |
| 23 | \( 1 + 5.97T + 23T^{2} \) |
| 29 | \( 1 - 3.52T + 29T^{2} \) |
| 31 | \( 1 - 10.8iT - 31T^{2} \) |
| 37 | \( 1 - 11.3iT - 37T^{2} \) |
| 41 | \( 1 + 7.26iT - 41T^{2} \) |
| 43 | \( 1 - 9.02T + 43T^{2} \) |
| 47 | \( 1 - 0.327T + 47T^{2} \) |
| 53 | \( 1 - 8.32T + 53T^{2} \) |
| 59 | \( 1 + 5.98iT - 59T^{2} \) |
| 61 | \( 1 + 13.2iT - 61T^{2} \) |
| 67 | \( 1 + 9.21T + 67T^{2} \) |
| 71 | \( 1 + 9.02T + 71T^{2} \) |
| 73 | \( 1 - 0.416T + 73T^{2} \) |
| 79 | \( 1 + 6.00iT - 79T^{2} \) |
| 83 | \( 1 - 2.62iT - 83T^{2} \) |
| 89 | \( 1 - 5.69iT - 89T^{2} \) |
| 97 | \( 1 - 6.21T + 97T^{2} \) |
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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
−9.454092912624929310815572297389, −8.937119328530567501077004088308, −8.242434292931117824119124206479, −6.99968491931453300785156333121, −6.34761714702167748689937147282, −5.74941573024765450064038358201, −4.94954492703861196309324316574, −3.09329537958339057265416548338, −2.12538052124947917273491902934, −1.30421075296655267456390784146,
0.954195254756989743222005301170, 2.10373991175011582878003600776, 2.77748733533652485464806967179, 4.23403349674510997079037252497, 5.75233090113995247634742428176, 5.96257400397150077089856570812, 7.16435266211672779773075932491, 7.75973987194222012214757236203, 8.876360122795568764429534807084, 9.507754871617039440651424454232