L(s) = 1 | + (0.866 + 0.5i)2-s + (0.866 − 1.5i)3-s + (0.499 + 0.866i)4-s + (1.5 − 0.866i)6-s + (1.73 + i)7-s + 0.999i·8-s + (−1.5 − 2.59i)9-s + (1.5 − 2.59i)11-s + 1.73·12-s + (1.73 − i)13-s + (0.999 + 1.73i)14-s + (−0.5 + 0.866i)16-s + 3i·17-s − 3i·18-s + 19-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (0.499 − 0.866i)3-s + (0.249 + 0.433i)4-s + (0.612 − 0.353i)6-s + (0.654 + 0.377i)7-s + 0.353i·8-s + (−0.5 − 0.866i)9-s + (0.452 − 0.783i)11-s + 0.499·12-s + (0.480 − 0.277i)13-s + (0.267 + 0.462i)14-s + (−0.125 + 0.216i)16-s + 0.727i·17-s − 0.707i·18-s + 0.229·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 + 0.232i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.972 + 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.40303 - 0.283034i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.40303 - 0.283034i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.866 + 1.5i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-1.73 - i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.73 + i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 3iT - 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 + (5.19 - 3i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 4iT - 37T^{2} \) |
| 41 | \( 1 + (4.5 + 7.79i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.866 - 0.5i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (5.19 + 3i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 12iT - 53T^{2} \) |
| 59 | \( 1 + (-1.5 - 2.59i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.33 - 2.5i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 - 11iT - 73T^{2} \) |
| 79 | \( 1 + (2 - 3.46i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (10.3 + 6i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + (-4.33 - 2.5i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.54634357293725921538241736585, −10.21328161457640303531678489000, −8.733703047672897402133758167910, −8.324903060608662562686265802093, −7.38332947979429658778190626789, −6.25029691053776191199681850829, −5.64503000377898697053221655777, −4.10147375122452446700351984605, −3.00538038939824399320404145252, −1.58053199738587509054199677009,
1.86232344382324101445466843707, 3.24948465149778468425840071771, 4.36981676749486436795081536550, 4.91058330644805171951861617830, 6.28542156010689608043724704067, 7.51375626212334676727221757919, 8.517911713700305473509675968795, 9.565730512945298119363828487064, 10.23031766895909532453799949936, 11.18097805283016277579210671583