| L(s) = 1 | + (−0.644 + 0.764i)2-s + (−0.981 − 0.192i)3-s + (−0.168 − 0.985i)4-s + (1.46 + 1.57i)5-s + (0.779 − 0.626i)6-s + (1.07 + 3.96i)7-s + (0.861 + 0.506i)8-s + (0.926 + 0.377i)9-s + (−2.14 + 0.103i)10-s + (0.0281 + 1.16i)11-s + (−0.0241 + 0.999i)12-s + (−1.93 − 0.0935i)13-s + (−3.72 − 1.72i)14-s + (−1.13 − 1.82i)15-s + (−0.943 + 0.331i)16-s + (−0.263 − 0.0787i)17-s + ⋯ |
| L(s) = 1 | + (−0.455 + 0.540i)2-s + (−0.566 − 0.110i)3-s + (−0.0841 − 0.492i)4-s + (0.654 + 0.703i)5-s + (0.318 − 0.255i)6-s + (0.407 + 1.49i)7-s + (0.304 + 0.179i)8-s + (0.308 + 0.125i)9-s + (−0.678 + 0.0328i)10-s + (0.00848 + 0.351i)11-s + (−0.00697 + 0.288i)12-s + (−0.536 − 0.0259i)13-s + (−0.994 − 0.462i)14-s + (−0.292 − 0.471i)15-s + (−0.235 + 0.0829i)16-s + (−0.0639 − 0.0190i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 786 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.784 - 0.620i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 786 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.784 - 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.321835 + 0.925332i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.321835 + 0.925332i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.644 - 0.764i)T \) |
| 3 | \( 1 + (0.981 + 0.192i)T \) |
| 131 | \( 1 + (-10.3 - 4.95i)T \) |
| good | 5 | \( 1 + (-1.46 - 1.57i)T + (-0.362 + 4.98i)T^{2} \) |
| 7 | \( 1 + (-1.07 - 3.96i)T + (-6.03 + 3.54i)T^{2} \) |
| 11 | \( 1 + (-0.0281 - 1.16i)T + (-10.9 + 0.531i)T^{2} \) |
| 13 | \( 1 + (1.93 + 0.0935i)T + (12.9 + 1.25i)T^{2} \) |
| 17 | \( 1 + (0.263 + 0.0787i)T + (14.2 + 9.31i)T^{2} \) |
| 19 | \( 1 + (-1.86 - 2.69i)T + (-6.73 + 17.7i)T^{2} \) |
| 23 | \( 1 + (-0.994 + 2.28i)T + (-15.6 - 16.8i)T^{2} \) |
| 29 | \( 1 + (4.60 - 1.87i)T + (20.7 - 20.2i)T^{2} \) |
| 31 | \( 1 + (-9.66 + 0.937i)T + (30.4 - 5.95i)T^{2} \) |
| 37 | \( 1 + (3.82 - 7.74i)T + (-22.4 - 29.3i)T^{2} \) |
| 41 | \( 1 + (6.84 + 2.40i)T + (31.9 + 25.6i)T^{2} \) |
| 43 | \( 1 + (-2.02 + 3.64i)T + (-22.6 - 36.5i)T^{2} \) |
| 47 | \( 1 + (9.43 - 2.32i)T + (41.6 - 21.8i)T^{2} \) |
| 53 | \( 1 + (4.05 - 2.94i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-2.98 - 3.91i)T + (-15.4 + 56.9i)T^{2} \) |
| 61 | \( 1 + (-1.20 - 3.71i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (-3.63 - 6.53i)T + (-35.3 + 56.9i)T^{2} \) |
| 71 | \( 1 + (1.09 - 0.968i)T + (8.55 - 70.4i)T^{2} \) |
| 73 | \( 1 + (4.24 - 13.0i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (1.58 + 13.0i)T + (-76.7 + 18.9i)T^{2} \) |
| 83 | \( 1 + (-4.65 + 0.679i)T + (79.5 - 23.7i)T^{2} \) |
| 89 | \( 1 + (5.46 + 3.96i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-6.96 + 4.56i)T + (38.7 - 88.9i)T^{2} \) |
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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
−10.29643051520871238348290270563, −9.916815980324062570274500448670, −8.875132004659178216775372560596, −8.096142031062359035473465679325, −7.00798410536546841537474379684, −6.26629699286038728254933374601, −5.52992326477132112347348539051, −4.74840422907274612480075368517, −2.80280093253513689527298577721, −1.76242592131297828881743054852,
0.63094485042444124672953515488, 1.71586354364658612743394928752, 3.42198207643825790657748223796, 4.57067266974286528958941132265, 5.24450582881451444893876853319, 6.57902099424572865803434229355, 7.44471005301717987926525575876, 8.309178564381315648870212002534, 9.465217145277852114540864938227, 9.897197917593831620871564281459
