L(s) = 1 | + 1.73·3-s + (−0.5 + 0.866i)5-s + (−1.73 − 2i)7-s + 2.99·9-s + (4.33 − 2.5i)11-s − 3.46i·13-s + (−0.866 + 1.49i)15-s + (−0.5 − 0.866i)17-s + (−0.866 − 0.5i)19-s + (−2.99 − 3.46i)21-s + (4.33 + 2.5i)23-s + (2 + 3.46i)25-s + 5.19·27-s + 3.46i·29-s + (7.79 − 4.5i)31-s + ⋯ |
L(s) = 1 | + 1.00·3-s + (−0.223 + 0.387i)5-s + (−0.654 − 0.755i)7-s + 0.999·9-s + (1.30 − 0.753i)11-s − 0.960i·13-s + (−0.223 + 0.387i)15-s + (−0.121 − 0.210i)17-s + (−0.198 − 0.114i)19-s + (−0.654 − 0.755i)21-s + (0.902 + 0.521i)23-s + (0.400 + 0.692i)25-s + 1.00·27-s + 0.643i·29-s + (1.39 − 0.808i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.00590 - 0.469905i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.00590 - 0.469905i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73T \) |
| 7 | \( 1 + (1.73 + 2i)T \) |
good | 5 | \( 1 + (0.5 - 0.866i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-4.33 + 2.5i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 3.46iT - 13T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.866 + 0.5i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.33 - 2.5i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 3.46iT - 29T^{2} \) |
| 31 | \( 1 + (-7.79 + 4.5i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.5 - 2.59i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 8T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + (-0.866 + 1.5i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (10.5 - 6.06i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.33 - 7.5i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (10.5 + 6.06i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.06 + 10.5i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 2iT - 71T^{2} \) |
| 73 | \( 1 + (-7.5 + 4.33i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.866 - 1.5i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 + (-6.5 + 11.2i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 3.46iT - 97T^{2} \) |
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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
−10.36630428050503016539948449467, −9.475442942118388321758708709555, −8.795762288897383122501619289229, −7.80776603522049181975625657125, −7.00052384979541171784723511517, −6.26058112726745435626685297025, −4.67484123865719220902953255915, −3.48121328455370505507698299020, −3.07842873938826525849999839885, −1.16921004664783307488677939992,
1.62154617051077850456520448401, 2.82274679960634504429462826564, 4.01733358191744714963154794214, 4.77290618329426056117389352783, 6.48493682941461070962935446647, 6.87374961567870506529979097335, 8.263853658239158054141587216049, 8.880635316733805105435871622907, 9.472053111216584380090179804947, 10.25865810487045676601876546194