| L(s) = 1 | + (1.30 − 0.544i)2-s + (−0.595 + 0.803i)3-s + (1.40 − 1.42i)4-s + (−1.97 + 2.18i)5-s + (−0.339 + 1.37i)6-s + (−0.674 − 1.26i)7-s + (1.05 − 2.62i)8-s + (−0.290 − 0.956i)9-s + (−1.39 + 3.92i)10-s + (1.26 + 5.03i)11-s + (0.304 + 1.97i)12-s + (−0.323 + 6.58i)13-s + (−1.56 − 1.27i)14-s + (−0.575 − 2.89i)15-s + (−0.0456 − 3.99i)16-s + (0.200 − 1.00i)17-s + ⋯ |
| L(s) = 1 | + (0.922 − 0.385i)2-s + (−0.343 + 0.463i)3-s + (0.703 − 0.711i)4-s + (−0.885 + 0.976i)5-s + (−0.138 + 0.560i)6-s + (−0.255 − 0.477i)7-s + (0.374 − 0.927i)8-s + (−0.0967 − 0.318i)9-s + (−0.440 + 1.24i)10-s + (0.380 + 1.51i)11-s + (0.0879 + 0.570i)12-s + (−0.0897 + 1.82i)13-s + (−0.419 − 0.342i)14-s + (−0.148 − 0.746i)15-s + (−0.0114 − 0.999i)16-s + (0.0486 − 0.244i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.148 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.148 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.27766 + 1.10022i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.27766 + 1.10022i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.30 + 0.544i)T \) |
| 3 | \( 1 + (0.595 - 0.803i)T \) |
| good | 5 | \( 1 + (1.97 - 2.18i)T + (-0.490 - 4.97i)T^{2} \) |
| 7 | \( 1 + (0.674 + 1.26i)T + (-3.88 + 5.82i)T^{2} \) |
| 11 | \( 1 + (-1.26 - 5.03i)T + (-9.70 + 5.18i)T^{2} \) |
| 13 | \( 1 + (0.323 - 6.58i)T + (-12.9 - 1.27i)T^{2} \) |
| 17 | \( 1 + (-0.200 + 1.00i)T + (-15.7 - 6.50i)T^{2} \) |
| 19 | \( 1 + (4.62 - 1.65i)T + (14.6 - 12.0i)T^{2} \) |
| 23 | \( 1 + (-0.488 - 0.595i)T + (-4.48 + 22.5i)T^{2} \) |
| 29 | \( 1 + (-2.95 - 4.93i)T + (-13.6 + 25.5i)T^{2} \) |
| 31 | \( 1 + (-1.59 + 0.662i)T + (21.9 - 21.9i)T^{2} \) |
| 37 | \( 1 + (0.0562 + 0.118i)T + (-23.4 + 28.6i)T^{2} \) |
| 41 | \( 1 + (0.298 - 3.03i)T + (-40.2 - 7.99i)T^{2} \) |
| 43 | \( 1 + (-3.00 + 2.22i)T + (12.4 - 41.1i)T^{2} \) |
| 47 | \( 1 + (-4.25 - 6.37i)T + (-17.9 + 43.4i)T^{2} \) |
| 53 | \( 1 + (3.73 - 6.22i)T + (-24.9 - 46.7i)T^{2} \) |
| 59 | \( 1 + (0.200 + 4.08i)T + (-58.7 + 5.78i)T^{2} \) |
| 61 | \( 1 + (13.4 - 1.98i)T + (58.3 - 17.7i)T^{2} \) |
| 67 | \( 1 + (2.10 + 14.2i)T + (-64.1 + 19.4i)T^{2} \) |
| 71 | \( 1 + (-12.8 - 3.89i)T + (59.0 + 39.4i)T^{2} \) |
| 73 | \( 1 + (3.53 - 6.61i)T + (-40.5 - 60.6i)T^{2} \) |
| 79 | \( 1 + (6.77 + 4.52i)T + (30.2 + 72.9i)T^{2} \) |
| 83 | \( 1 + (-4.80 + 10.1i)T + (-52.6 - 64.1i)T^{2} \) |
| 89 | \( 1 + (3.12 - 3.80i)T + (-17.3 - 87.2i)T^{2} \) |
| 97 | \( 1 + (0.257 + 0.622i)T + (-68.5 + 68.5i)T^{2} \) |
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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
−10.72976568661569350222359445118, −9.974182789852111754332041545528, −9.149379077008202060189228852505, −7.42622446696796025780244371486, −6.89531671578196245798869041313, −6.24561252937528370828193081238, −4.54344769608380942571553503472, −4.31866366901487609368259599633, −3.27312940183156900764587028568, −1.90989911095957337229572319174,
0.65362204469047722811109338815, 2.71027044177220478456770477903, 3.74255239526243331522177102197, 4.81709429561256987852955197938, 5.74679741411309346851977335032, 6.28582006805046064549022305245, 7.58534546097761834150390985026, 8.325051231267614601631799170763, 8.725267717814560149432627068951, 10.49505188812314464106352426076
