L(s) = 1 | + (−2.75 + 1.58i)3-s + (0.5 − 0.866i)5-s + (−1.04 − 2.43i)7-s + (3.55 − 6.15i)9-s + (−1.21 − 2.09i)11-s + 1.53·13-s + 3.17i·15-s + (−6.58 + 3.79i)17-s + (1.52 + 0.883i)19-s + (6.73 + 5.03i)21-s + (5.66 + 3.26i)23-s + (−0.499 − 0.866i)25-s + 13.0i·27-s + 2.34i·29-s + (−1.04 − 1.80i)31-s + ⋯ |
L(s) = 1 | + (−1.58 + 0.917i)3-s + (0.223 − 0.387i)5-s + (−0.394 − 0.919i)7-s + (1.18 − 2.05i)9-s + (−0.364 − 0.631i)11-s + 0.426·13-s + 0.820i·15-s + (−1.59 + 0.921i)17-s + (0.350 + 0.202i)19-s + (1.47 + 1.09i)21-s + (1.18 + 0.681i)23-s + (−0.0999 − 0.173i)25-s + 2.51i·27-s + 0.435i·29-s + (−0.187 − 0.323i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.892 - 0.450i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.892 - 0.450i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2161509618\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2161509618\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (1.04 + 2.43i)T \) |
good | 3 | \( 1 + (2.75 - 1.58i)T + (1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (1.21 + 2.09i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 1.53T + 13T^{2} \) |
| 17 | \( 1 + (6.58 - 3.79i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.52 - 0.883i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.66 - 3.26i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 2.34iT - 29T^{2} \) |
| 31 | \( 1 + (1.04 + 1.80i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.47 + 1.42i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 6.90iT - 41T^{2} \) |
| 43 | \( 1 - 1.39T + 43T^{2} \) |
| 47 | \( 1 + (5.65 - 9.79i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (6.82 - 3.93i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (4.90 - 2.82i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4.93 - 8.54i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3.13 + 5.42i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 3.98iT - 71T^{2} \) |
| 73 | \( 1 + (2.73 - 1.57i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (1.75 + 1.01i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 0.288iT - 83T^{2} \) |
| 89 | \( 1 + (-12.0 - 6.95i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 0.249iT - 97T^{2} \) |
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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
−10.59837356420594564202208523633, −9.456975049709599533368698748788, −8.897587860724418296371246249700, −7.50753580173378712426782894232, −6.49132783223869500988926112770, −5.95992088003920386612971472677, −5.01999662130654038851659741381, −4.27754324955388466221293151615, −3.41333720856158831496064859523, −1.20288580340881391622831758614,
0.12939945024878168940900039808, 1.79682862851941396266763527928, 2.81277647266881284690783476697, 4.74421269280290802884271682976, 5.23600744397047068797805256677, 6.37182667321712264463088447115, 6.65055157326359355358515818079, 7.47677407591278386938354516811, 8.652463964873957283181308843164, 9.611420563609552661614370484269