L(s) = 1 | + (1.28 − 1.16i)3-s + (1.69 + 1.69i)5-s + (1.36 − 0.366i)7-s + (0.292 − 2.98i)9-s + (1.69 + 0.453i)11-s + (−1.59 + 3.23i)13-s + (4.14 + 0.202i)15-s + (1.07 − 1.85i)17-s + (0.267 + i)19-s + (1.32 − 2.05i)21-s + 0.732i·25-s + (−3.09 − 4.17i)27-s + (4.79 − 2.76i)29-s + (−4.46 + 4.46i)31-s + (2.69 − 1.38i)33-s + ⋯ |
L(s) = 1 | + (0.740 − 0.671i)3-s + (0.757 + 0.757i)5-s + (0.516 − 0.138i)7-s + (0.0975 − 0.995i)9-s + (0.510 + 0.136i)11-s + (−0.443 + 0.896i)13-s + (1.06 + 0.0523i)15-s + (0.260 − 0.450i)17-s + (0.0614 + 0.229i)19-s + (0.289 − 0.449i)21-s + 0.146i·25-s + (−0.596 − 0.802i)27-s + (0.889 − 0.513i)29-s + (−0.801 + 0.801i)31-s + (0.470 − 0.241i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 + 0.268i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.963 + 0.268i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.23966 - 0.306549i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.23966 - 0.306549i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.28 + 1.16i)T \) |
| 13 | \( 1 + (1.59 - 3.23i)T \) |
good | 5 | \( 1 + (-1.69 - 1.69i)T + 5iT^{2} \) |
| 7 | \( 1 + (-1.36 + 0.366i)T + (6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (-1.69 - 0.453i)T + (9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + (-1.07 + 1.85i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.267 - i)T + (-16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.79 + 2.76i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (4.46 - 4.46i)T - 31iT^{2} \) |
| 37 | \( 1 + (1.76 - 6.59i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (0.166 - 0.619i)T + (-35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (7.09 + 4.09i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-6.77 + 6.77i)T - 47iT^{2} \) |
| 53 | \( 1 + 4.62iT - 53T^{2} \) |
| 59 | \( 1 + (1.23 + 4.62i)T + (-51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-8.46 - 2.26i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (4.62 - 1.23i)T + (61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (6.09 + 6.09i)T + 73iT^{2} \) |
| 79 | \( 1 + 2T + 79T^{2} \) |
| 83 | \( 1 + (-1.23 - 1.23i)T + 83iT^{2} \) |
| 89 | \( 1 + (9.70 + 2.60i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (-3.36 - 12.5i)T + (-84.0 + 48.5i)T^{2} \) |
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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
−10.38106697208301844202661360169, −9.675926689006692865745692247545, −8.830356615641540883239295859192, −7.897034025897756341533857261323, −6.87136866341011276503099578471, −6.48154660441220887355035311408, −5.06536397895484068475810801163, −3.72044108469972944741742403507, −2.52865162578355375189866279071, −1.57482619999457212044462351694,
1.55481508596812959132166508044, 2.83573064122903657920032121676, 4.11368456079074009281044834302, 5.11077643778160358064101348762, 5.79620636917876864939049069327, 7.33121631600401117175524381193, 8.278075875193002467978717603509, 8.946960074085322691248004718331, 9.676704957507271011621732163033, 10.43759439067414548438804868406