L(s) = 1 | + (1.70 − 0.292i)3-s + (−0.707 + 2.12i)5-s + (1 + i)7-s + (2.82 − i)9-s + 1.41i·11-s + (−0.585 + 3.82i)15-s + (1.41 − 1.41i)17-s − 4i·19-s + (2 + 1.41i)21-s + (−2.82 − 2.82i)23-s + (−3.99 − 3i)25-s + (4.53 − 2.53i)27-s − 7.07·29-s + 2·31-s + (0.414 + 2.41i)33-s + ⋯ |
L(s) = 1 | + (0.985 − 0.169i)3-s + (−0.316 + 0.948i)5-s + (0.377 + 0.377i)7-s + (0.942 − 0.333i)9-s + 0.426i·11-s + (−0.151 + 0.988i)15-s + (0.342 − 0.342i)17-s − 0.917i·19-s + (0.436 + 0.308i)21-s + (−0.589 − 0.589i)23-s + (−0.799 − 0.600i)25-s + (0.872 − 0.487i)27-s − 1.31·29-s + 0.359·31-s + (0.0721 + 0.420i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.391i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.920 - 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.61316 + 0.328467i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.61316 + 0.328467i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.70 + 0.292i)T \) |
| 5 | \( 1 + (0.707 - 2.12i)T \) |
good | 7 | \( 1 + (-1 - i)T + 7iT^{2} \) |
| 11 | \( 1 - 1.41iT - 11T^{2} \) |
| 13 | \( 1 - 13iT^{2} \) |
| 17 | \( 1 + (-1.41 + 1.41i)T - 17iT^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + (2.82 + 2.82i)T + 23iT^{2} \) |
| 29 | \( 1 + 7.07T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + (-6 - 6i)T + 37iT^{2} \) |
| 41 | \( 1 + 5.65iT - 41T^{2} \) |
| 43 | \( 1 + (6 - 6i)T - 43iT^{2} \) |
| 47 | \( 1 - 47iT^{2} \) |
| 53 | \( 1 + (2.82 + 2.82i)T + 53iT^{2} \) |
| 59 | \( 1 + 9.89T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + (-4 - 4i)T + 67iT^{2} \) |
| 71 | \( 1 + 14.1iT - 71T^{2} \) |
| 73 | \( 1 + (5 - 5i)T - 73iT^{2} \) |
| 79 | \( 1 + 6iT - 79T^{2} \) |
| 83 | \( 1 + (-8.48 - 8.48i)T + 83iT^{2} \) |
| 89 | \( 1 - 2.82T + 89T^{2} \) |
| 97 | \( 1 + (-3 - 3i)T + 97iT^{2} \) |
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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
−12.17357955919665420545723235917, −11.27231362804532560212750937811, −10.15896183076687530651832536691, −9.276427151016199512669633371663, −8.138030654360897630184246494722, −7.38028564105894202736701374826, −6.37190787957464485731840504226, −4.64530619211594333345999103072, −3.31221585966140152183897399587, −2.18348590428698455706011520867,
1.61753522619537926076848143016, 3.52772229754449689331233736598, 4.45151706131114117735597423237, 5.79013113471455402614509553785, 7.57450160831616388846090142798, 8.087185093384237320099461870712, 9.086700700392381725424574911571, 9.933370045323478332606324401378, 11.09003167545535410593262849981, 12.22870443400751744384377365327