L(s) = 1 | + (−0.707 + 1.22i)2-s + (1.5 − 0.866i)3-s + (−0.999 − 1.73i)4-s + 2.44i·6-s + (5.26 − 4.61i)7-s + 2.82·8-s + (1.5 − 2.59i)9-s + (5.41 + 9.37i)11-s + (−2.99 − 1.73i)12-s − 19.2i·13-s + (1.93 + 9.70i)14-s + (−2.00 + 3.46i)16-s + (8.89 − 5.13i)17-s + (2.12 + 3.67i)18-s + (−18.0 − 10.4i)19-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.5 − 0.288i)3-s + (−0.249 − 0.433i)4-s + 0.408i·6-s + (0.751 − 0.659i)7-s + 0.353·8-s + (0.166 − 0.288i)9-s + (0.491 + 0.852i)11-s + (−0.249 − 0.144i)12-s − 1.48i·13-s + (0.137 + 0.693i)14-s + (−0.125 + 0.216i)16-s + (0.523 − 0.302i)17-s + (0.117 + 0.204i)18-s + (−0.951 − 0.549i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.558 + 0.829i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.558 + 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.868927454\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.868927454\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 1.22i)T \) |
| 3 | \( 1 + (-1.5 + 0.866i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-5.26 + 4.61i)T \) |
good | 11 | \( 1 + (-5.41 - 9.37i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + 19.2iT - 169T^{2} \) |
| 17 | \( 1 + (-8.89 + 5.13i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (18.0 + 10.4i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-10.5 + 18.2i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 - 19.0T + 841T^{2} \) |
| 31 | \( 1 + (34.6 - 20.0i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (25.1 - 43.6i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + 22.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 48.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + (57.6 + 33.2i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-2.47 - 4.28i)T + (-1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (24.4 - 14.0i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (60.6 + 35.0i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-9.65 - 16.7i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 49.4T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-115. + 66.4i)T + (2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-45.0 + 78.0i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + 101. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (-34.3 - 19.8i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 - 68.6iT - 9.40e3T^{2} \) |
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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
−9.438375659902204633542168950321, −8.550239000898949138907342905779, −7.939094574907414244087236559495, −7.18822127080248374154043188355, −6.51625656947097895481025536909, −5.22260411962976726604838242162, −4.49922562016579088381535174400, −3.24594950502401660083108739897, −1.84336854533919440983517092078, −0.64004500570792864264895198614,
1.43203077918030866373425686205, 2.29172482623205313381396046389, 3.56640823100554485548931873239, 4.30149528042787568924013297218, 5.44447168833141857684548259507, 6.49611588906431701325542283116, 7.69971264725812959110010255556, 8.420097085260941157108595028553, 9.126698651410750970897937172478, 9.582044219434662221359562402243