L(s) = 1 | + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + 1.73·5-s + (0.5 − 2.59i)7-s + 0.999i·8-s + (1.49 + 0.866i)10-s + (1.73 − 2i)14-s + (−0.5 + 0.866i)16-s + (1.73 − 3i)17-s + (6 − 3.46i)19-s + (0.866 + 1.49i)20-s + 6i·23-s − 2.00·25-s + (2.5 − 0.866i)28-s + (7.79 − 4.5i)29-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + 0.774·5-s + (0.188 − 0.981i)7-s + 0.353i·8-s + (0.474 + 0.273i)10-s + (0.462 − 0.534i)14-s + (−0.125 + 0.216i)16-s + (0.420 − 0.727i)17-s + (1.37 − 0.794i)19-s + (0.193 + 0.335i)20-s + 1.25i·23-s − 0.400·25-s + (0.472 − 0.163i)28-s + (1.44 − 0.835i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.110i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.110i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.795533612\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.795533612\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.5 + 2.59i)T \) |
good | 5 | \( 1 - 1.73T + 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.73 + 3i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-6 + 3.46i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + (-7.79 + 4.5i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (3 - 1.73i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2 + 3.46i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.73 - 3i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-4 - 6.92i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.73 + 3i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.59 - 1.5i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-6.06 - 10.5i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3 + 1.73i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (7 + 12.1i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 6iT - 71T^{2} \) |
| 73 | \( 1 + (10.5 + 6.06i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5.5 - 9.52i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-8.66 - 15i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-5.19 - 9i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-6 + 3.46i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.780379990646521993993058757304, −9.165489438207780886426561409362, −7.82031477277873499258537374449, −7.35176069053409261721157146276, −6.43833051108593193571865909884, −5.48816446748388282156307967318, −4.80682614768122444897990905927, −3.70296973349955313367848209947, −2.69575563735575496139580642784, −1.20161706674668295012847074045,
1.46923222089454465193784103114, 2.47879456273766357897552518537, 3.46626130918610480432785189005, 4.72256886851633731956872472458, 5.62489608776219843822928289588, 6.05093445852663352246468487451, 7.16179633061442411212625238381, 8.323848127501314230610520633866, 9.035041131724654924214461402758, 10.05537507804210476656021406734