| L(s) = 1 | + (−1.38 − 0.272i)2-s + (−0.703 − 2.75i)3-s + (1.85 + 0.755i)4-s + (−3.43 + 0.747i)5-s + (0.225 + 4.01i)6-s + (−0.125 + 0.694i)7-s + (−2.36 − 1.55i)8-s + (−4.47 + 2.44i)9-s + (4.96 − 0.101i)10-s + (1.78 + 2.78i)11-s + (0.781 − 5.63i)12-s + (−4.92 − 2.92i)13-s + (0.363 − 0.929i)14-s + (4.47 + 8.94i)15-s + (2.85 + 2.79i)16-s + (−0.135 + 1.89i)17-s + ⋯ |
| L(s) = 1 | + (−0.981 − 0.192i)2-s + (−0.406 − 1.59i)3-s + (0.925 + 0.377i)4-s + (−1.53 + 0.334i)5-s + (0.0920 + 1.64i)6-s + (−0.0473 + 0.262i)7-s + (−0.835 − 0.549i)8-s + (−1.49 + 0.814i)9-s + (1.57 − 0.0320i)10-s + (0.539 + 0.839i)11-s + (0.225 − 1.62i)12-s + (−1.36 − 0.810i)13-s + (0.0970 − 0.248i)14-s + (1.15 + 2.30i)15-s + (0.714 + 0.699i)16-s + (−0.0328 + 0.459i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.941 + 0.336i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.941 + 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.421636 - 0.0730432i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.421636 - 0.0730432i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.38 + 0.272i)T \) |
| 89 | \( 1 + (5.20 - 7.87i)T \) |
| good | 3 | \( 1 + (0.703 + 2.75i)T + (-2.63 + 1.43i)T^{2} \) |
| 5 | \( 1 + (3.43 - 0.747i)T + (4.54 - 2.07i)T^{2} \) |
| 7 | \( 1 + (0.125 - 0.694i)T + (-6.55 - 2.44i)T^{2} \) |
| 11 | \( 1 + (-1.78 - 2.78i)T + (-4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (4.92 + 2.92i)T + (6.23 + 11.4i)T^{2} \) |
| 17 | \( 1 + (0.135 - 1.89i)T + (-16.8 - 2.41i)T^{2} \) |
| 19 | \( 1 + (-3.37 + 4.19i)T + (-4.03 - 18.5i)T^{2} \) |
| 23 | \( 1 + (-0.523 - 4.87i)T + (-22.4 + 4.88i)T^{2} \) |
| 29 | \( 1 + (-3.75 - 5.40i)T + (-10.1 + 27.1i)T^{2} \) |
| 31 | \( 1 + (0.00905 - 0.0841i)T + (-30.2 - 6.58i)T^{2} \) |
| 37 | \( 1 + (-5.94 - 2.46i)T + (26.1 + 26.1i)T^{2} \) |
| 41 | \( 1 + (-2.26 + 1.34i)T + (19.6 - 35.9i)T^{2} \) |
| 43 | \( 1 + (7.99 + 5.55i)T + (15.0 + 40.2i)T^{2} \) |
| 47 | \( 1 + (2.35 + 1.76i)T + (13.2 + 45.0i)T^{2} \) |
| 53 | \( 1 + (1.44 + 1.92i)T + (-14.9 + 50.8i)T^{2} \) |
| 59 | \( 1 + (-5.50 - 1.40i)T + (51.7 + 28.2i)T^{2} \) |
| 61 | \( 1 + (-0.521 + 14.5i)T + (-60.8 - 4.35i)T^{2} \) |
| 67 | \( 1 + (1.39 - 9.66i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (-0.146 + 0.671i)T + (-64.5 - 29.4i)T^{2} \) |
| 73 | \( 1 + (0.803 + 2.73i)T + (-61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (-7.80 - 4.25i)T + (42.7 + 66.4i)T^{2} \) |
| 83 | \( 1 + (-2.85 + 5.70i)T + (-49.7 - 66.4i)T^{2} \) |
| 97 | \( 1 + (-16.3 - 10.4i)T + (40.2 + 88.2i)T^{2} \) |
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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
−10.52215317100558788872985495876, −9.430242477735247165469685006708, −8.318166057081587871480412881257, −7.59282052499495983420882552636, −7.21379194217769395031671882014, −6.55758928579886307924736978755, −5.10387074424458534556889956974, −3.38003303777087407811706935943, −2.29649954782189196720660126379, −0.842202662199443080157276750648,
0.47257624930168099352229059089, 3.02401950616094740519349382799, 4.11457597148884472957149356389, 4.81802702268610838311270101844, 6.04521069090313565402129850496, 7.20778246220985876099669720096, 8.077942846507163580713194998575, 8.879245038042553329365904515324, 9.676347657631926995403424734367, 10.26955759320704128757671410068
