| L(s) = 1 | + (1.10 + 0.888i)2-s + (1.24 − 1.50i)3-s + (0.422 + 1.95i)4-s + (−0.0625 − 2.23i)5-s + (2.70 − 0.549i)6-s + (2.70 + 0.878i)7-s + (−1.27 + 2.52i)8-s + (−0.149 − 0.784i)9-s + (1.91 − 2.51i)10-s + (0.494 + 3.91i)11-s + (3.46 + 1.79i)12-s + (0.919 + 4.82i)13-s + (2.19 + 3.36i)14-s + (−3.43 − 2.68i)15-s + (−3.64 + 1.65i)16-s + (0.499 − 1.94i)17-s + ⋯ |
| L(s) = 1 | + (0.778 + 0.628i)2-s + (0.717 − 0.867i)3-s + (0.211 + 0.977i)4-s + (−0.0279 − 0.999i)5-s + (1.10 − 0.224i)6-s + (1.02 + 0.332i)7-s + (−0.449 + 0.893i)8-s + (−0.0499 − 0.261i)9-s + (0.605 − 0.795i)10-s + (0.149 + 1.17i)11-s + (0.999 + 0.517i)12-s + (0.255 + 1.33i)13-s + (0.586 + 0.900i)14-s + (−0.886 − 0.692i)15-s + (−0.910 + 0.412i)16-s + (0.121 − 0.472i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 - 0.386i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.922 - 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.37755 + 0.679299i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.37755 + 0.679299i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.10 - 0.888i)T \) |
| 5 | \( 1 + (0.0625 + 2.23i)T \) |
| good | 3 | \( 1 + (-1.24 + 1.50i)T + (-0.562 - 2.94i)T^{2} \) |
| 7 | \( 1 + (-2.70 - 0.878i)T + (5.66 + 4.11i)T^{2} \) |
| 11 | \( 1 + (-0.494 - 3.91i)T + (-10.6 + 2.73i)T^{2} \) |
| 13 | \( 1 + (-0.919 - 4.82i)T + (-12.0 + 4.78i)T^{2} \) |
| 17 | \( 1 + (-0.499 + 1.94i)T + (-14.8 - 8.18i)T^{2} \) |
| 19 | \( 1 + (-4.73 + 3.91i)T + (3.56 - 18.6i)T^{2} \) |
| 23 | \( 1 + (4.66 + 4.96i)T + (-1.44 + 22.9i)T^{2} \) |
| 29 | \( 1 + (-2.75 - 0.173i)T + (28.7 + 3.63i)T^{2} \) |
| 31 | \( 1 + (7.19 + 1.84i)T + (27.1 + 14.9i)T^{2} \) |
| 37 | \( 1 + (-5.26 + 2.89i)T + (19.8 - 31.2i)T^{2} \) |
| 41 | \( 1 + (0.720 + 0.676i)T + (2.57 + 40.9i)T^{2} \) |
| 43 | \( 1 + (4.55 + 3.31i)T + (13.2 + 40.8i)T^{2} \) |
| 47 | \( 1 + (-3.72 + 9.41i)T + (-34.2 - 32.1i)T^{2} \) |
| 53 | \( 1 + (1.03 + 1.63i)T + (-22.5 + 47.9i)T^{2} \) |
| 59 | \( 1 + (-6.34 - 2.98i)T + (37.6 + 45.4i)T^{2} \) |
| 61 | \( 1 + (3.27 + 3.49i)T + (-3.83 + 60.8i)T^{2} \) |
| 67 | \( 1 + (-0.775 - 12.3i)T + (-66.4 + 8.39i)T^{2} \) |
| 71 | \( 1 + (10.9 + 4.33i)T + (51.7 + 48.6i)T^{2} \) |
| 73 | \( 1 + (6.14 - 2.89i)T + (46.5 - 56.2i)T^{2} \) |
| 79 | \( 1 + (10.6 - 12.8i)T + (-14.8 - 77.6i)T^{2} \) |
| 83 | \( 1 + (-2.98 - 3.60i)T + (-15.5 + 81.5i)T^{2} \) |
| 89 | \( 1 + (0.929 + 1.97i)T + (-56.7 + 68.5i)T^{2} \) |
| 97 | \( 1 + (2.74 + 0.172i)T + (96.2 + 12.1i)T^{2} \) |
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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
−9.653426458019372126766021089681, −8.748306110502611293485085831014, −8.327234516337968659848834364346, −7.32978897582589727299805096649, −6.97260463978261664505879853029, −5.57492640665259279592544825387, −4.75721774608362439764073468180, −4.11786654911652475567509386006, −2.41163106112957579407370669154, −1.69873303920350591282792159884,
1.42266096783668394116389254366, 3.07010348070351065303204036867, 3.39160171101150717260900993313, 4.29163301931955485092247035568, 5.54308442058279035969976774767, 6.11356039453423043503080777380, 7.59561048799199612636666397779, 8.218962267232715133091951313273, 9.415917331939092937766980050162, 10.18091071227267523754313958036
