L(s) = 1 | + (−0.636 − 0.636i)3-s + (1.68 − 1.47i)5-s + (−0.707 + 0.707i)7-s − 2.19i·9-s + 0.0126i·11-s + (1.39 − 1.39i)13-s + (−2.00 − 0.134i)15-s + (−1.83 − 1.83i)17-s + 1.25·19-s + 0.899·21-s + (−0.761 − 0.761i)23-s + (0.666 − 4.95i)25-s + (−3.30 + 3.30i)27-s + 1.27i·29-s − 8.86i·31-s + ⋯ |
L(s) = 1 | + (−0.367 − 0.367i)3-s + (0.752 − 0.658i)5-s + (−0.267 + 0.267i)7-s − 0.730i·9-s + 0.00380i·11-s + (0.386 − 0.386i)13-s + (−0.518 − 0.0346i)15-s + (−0.444 − 0.444i)17-s + 0.288·19-s + 0.196·21-s + (−0.158 − 0.158i)23-s + (0.133 − 0.991i)25-s + (−0.635 + 0.635i)27-s + 0.237i·29-s − 1.59i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0368 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0368 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.902501 - 0.936391i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.902501 - 0.936391i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.68 + 1.47i)T \) |
| 7 | \( 1 + (0.707 - 0.707i)T \) |
good | 3 | \( 1 + (0.636 + 0.636i)T + 3iT^{2} \) |
| 11 | \( 1 - 0.0126iT - 11T^{2} \) |
| 13 | \( 1 + (-1.39 + 1.39i)T - 13iT^{2} \) |
| 17 | \( 1 + (1.83 + 1.83i)T + 17iT^{2} \) |
| 19 | \( 1 - 1.25T + 19T^{2} \) |
| 23 | \( 1 + (0.761 + 0.761i)T + 23iT^{2} \) |
| 29 | \( 1 - 1.27iT - 29T^{2} \) |
| 31 | \( 1 + 8.86iT - 31T^{2} \) |
| 37 | \( 1 + (0.649 + 0.649i)T + 37iT^{2} \) |
| 41 | \( 1 - 5.52T + 41T^{2} \) |
| 43 | \( 1 + (4.70 + 4.70i)T + 43iT^{2} \) |
| 47 | \( 1 + (-2.07 + 2.07i)T - 47iT^{2} \) |
| 53 | \( 1 + (2.69 - 2.69i)T - 53iT^{2} \) |
| 59 | \( 1 - 0.236T + 59T^{2} \) |
| 61 | \( 1 - 12.3T + 61T^{2} \) |
| 67 | \( 1 + (4.86 - 4.86i)T - 67iT^{2} \) |
| 71 | \( 1 + 12.9iT - 71T^{2} \) |
| 73 | \( 1 + (4.87 - 4.87i)T - 73iT^{2} \) |
| 79 | \( 1 - 10.0T + 79T^{2} \) |
| 83 | \( 1 + (-12.2 - 12.2i)T + 83iT^{2} \) |
| 89 | \( 1 - 8.25iT - 89T^{2} \) |
| 97 | \( 1 + (-4.81 - 4.81i)T + 97iT^{2} \) |
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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
−10.49487928469039157150226907813, −9.491891245510463309137931132829, −8.999880788315812040350461541303, −7.88847166092531967606851278577, −6.68706989188256976143645325911, −5.97380737156747401999994212616, −5.15544961276251213449735623122, −3.81077726180133843141616181067, −2.32125715743123041007343943469, −0.799892612061275608346684821512,
1.82901630497717202755301688976, 3.17169251530805870578072012883, 4.45167675264081188788593648218, 5.51724783526132760709828112830, 6.38728560797327909655232153063, 7.24147508040200217375309027084, 8.392058712992133865396431815776, 9.461881772634224199173458709177, 10.23896914399086583997849885134, 10.84688260495090118828613589984