L(s) = 1 | + (−1.35 − 0.364i)3-s + (−2.13 + 0.666i)5-s + (2.59 + 0.501i)7-s + (−0.882 − 0.509i)9-s + (1.86 + 3.23i)11-s + (−4.55 + 4.55i)13-s + (3.14 − 0.129i)15-s + (−1.58 + 5.91i)17-s + (−0.616 + 1.06i)19-s + (−3.34 − 1.62i)21-s + (−2.69 + 0.721i)23-s + (4.11 − 2.84i)25-s + (3.99 + 3.99i)27-s + 2.82i·29-s + (5.94 − 3.43i)31-s + ⋯ |
L(s) = 1 | + (−0.784 − 0.210i)3-s + (−0.954 + 0.298i)5-s + (0.981 + 0.189i)7-s + (−0.294 − 0.169i)9-s + (0.563 + 0.975i)11-s + (−1.26 + 1.26i)13-s + (0.811 − 0.0333i)15-s + (−0.384 + 1.43i)17-s + (−0.141 + 0.244i)19-s + (−0.730 − 0.355i)21-s + (−0.561 + 0.150i)23-s + (0.822 − 0.569i)25-s + (0.769 + 0.769i)27-s + 0.525i·29-s + (1.06 − 0.616i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.215 - 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.215 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.383685 + 0.477629i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.383685 + 0.477629i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (2.13 - 0.666i)T \) |
| 7 | \( 1 + (-2.59 - 0.501i)T \) |
good | 3 | \( 1 + (1.35 + 0.364i)T + (2.59 + 1.5i)T^{2} \) |
| 11 | \( 1 + (-1.86 - 3.23i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (4.55 - 4.55i)T - 13iT^{2} \) |
| 17 | \( 1 + (1.58 - 5.91i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (0.616 - 1.06i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.69 - 0.721i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 - 2.82iT - 29T^{2} \) |
| 31 | \( 1 + (-5.94 + 3.43i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2.01 + 7.52i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + 5.56iT - 41T^{2} \) |
| 43 | \( 1 + (1.95 + 1.95i)T + 43iT^{2} \) |
| 47 | \( 1 + (11.1 - 2.98i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (2.53 - 9.44i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (0.916 + 1.58i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.43 + 1.98i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.93 - 2.12i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 - 0.570T + 71T^{2} \) |
| 73 | \( 1 + (2.03 + 0.546i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-9.47 - 5.47i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-8.80 + 8.80i)T - 83iT^{2} \) |
| 89 | \( 1 + (3.00 - 5.20i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.70 - 3.70i)T + 97iT^{2} \) |
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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
−12.08605042738754502582224071925, −11.38895717853583820446534036821, −10.54616820373705040462014751447, −9.236131351706964519166000331298, −8.151646383231464712789098324647, −7.16856706759077226186639027343, −6.30607201154255947955869654862, −4.86947171551509003708787449709, −4.04332507734869933223641459583, −1.96854755820331507335958124198,
0.50888744727726992361461133672, 3.01318990091861485766580687487, 4.71066923287425649686244432964, 5.14697953168209659939518457937, 6.61952406672088782470777026032, 7.908730446444391653473948593096, 8.416516767332709150386270226914, 9.882877179274273785321257504485, 10.94712074710233610258016529880, 11.65065884236879887896053125695