L(s) = 1 | − 0.387i·2-s − 0.494i·3-s + 1.84·4-s + (−1.95 + 1.08i)5-s − 0.191·6-s + 2.37i·7-s − 1.49i·8-s + 2.75·9-s + (0.421 + 0.756i)10-s − 11-s − 0.914i·12-s + 0.475i·13-s + 0.919·14-s + (0.538 + 0.965i)15-s + 3.12·16-s + 0.596i·17-s + ⋯ |
L(s) = 1 | − 0.273i·2-s − 0.285i·3-s + 0.924·4-s + (−0.873 + 0.486i)5-s − 0.0781·6-s + 0.896i·7-s − 0.527i·8-s + 0.918·9-s + (0.133 + 0.239i)10-s − 0.301·11-s − 0.263i·12-s + 0.131i·13-s + 0.245·14-s + (0.138 + 0.249i)15-s + 0.780·16-s + 0.144i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.873 - 0.486i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.873 - 0.486i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.776913252\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.776913252\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (1.95 - 1.08i)T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 + T \) |
good | 2 | \( 1 + 0.387iT - 2T^{2} \) |
| 3 | \( 1 + 0.494iT - 3T^{2} \) |
| 7 | \( 1 - 2.37iT - 7T^{2} \) |
| 13 | \( 1 - 0.475iT - 13T^{2} \) |
| 17 | \( 1 - 0.596iT - 17T^{2} \) |
| 23 | \( 1 - 8.79iT - 23T^{2} \) |
| 29 | \( 1 + 3.72T + 29T^{2} \) |
| 31 | \( 1 - 7.08T + 31T^{2} \) |
| 37 | \( 1 - 8.28iT - 37T^{2} \) |
| 41 | \( 1 - 6.65T + 41T^{2} \) |
| 43 | \( 1 - 10.7iT - 43T^{2} \) |
| 47 | \( 1 - 3.68iT - 47T^{2} \) |
| 53 | \( 1 + 1.59iT - 53T^{2} \) |
| 59 | \( 1 - 7.09T + 59T^{2} \) |
| 61 | \( 1 + 6.32T + 61T^{2} \) |
| 67 | \( 1 + 10.2iT - 67T^{2} \) |
| 71 | \( 1 + 11.0T + 71T^{2} \) |
| 73 | \( 1 + 12.0iT - 73T^{2} \) |
| 79 | \( 1 - 16.5T + 79T^{2} \) |
| 83 | \( 1 + 10.5iT - 83T^{2} \) |
| 89 | \( 1 + 7.65T + 89T^{2} \) |
| 97 | \( 1 + 3.30iT - 97T^{2} \) |
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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
−10.10330930421270656613657770304, −9.298946717864302655270976019641, −7.953287976500856377957000668373, −7.61389246004769758928672450413, −6.64924242990514147723786501875, −5.97267764168531691631895374464, −4.64012220190335935104776737929, −3.48958313508395158225522777277, −2.61629223567932070810128818464, −1.45724806949068192598537227094,
0.861274610707743954458580422574, 2.43074445902524973203137801936, 3.82256068863662557324947540281, 4.44245320949356006298597400331, 5.53591984232612148304344443707, 6.79216562546406367838848599078, 7.27235869552280282263731440072, 8.020369774730017465238359140419, 8.862400536724500830778214463132, 10.12828810358082744605747478261