| L(s) = 1 | + (0.927 − 1.06i)2-s + (0.306 + 1.73i)3-s + (−0.280 − 1.98i)4-s + (−0.220 + 0.184i)5-s + (2.14 + 1.28i)6-s + (−0.588 − 0.339i)7-s + (−2.37 − 1.53i)8-s + (−0.110 + 0.0403i)9-s + (−0.00686 + 0.406i)10-s + (−3.85 + 2.22i)11-s + (3.35 − 1.09i)12-s + (−3.41 − 0.602i)13-s + (−0.908 + 0.313i)14-s + (−0.388 − 0.326i)15-s + (−3.84 + 1.11i)16-s + (4.15 + 1.51i)17-s + ⋯ |
| L(s) = 1 | + (0.655 − 0.755i)2-s + (0.177 + 1.00i)3-s + (−0.140 − 0.990i)4-s + (−0.0984 + 0.0826i)5-s + (0.874 + 0.524i)6-s + (−0.222 − 0.128i)7-s + (−0.839 − 0.543i)8-s + (−0.0369 + 0.0134i)9-s + (−0.00217 + 0.128i)10-s + (−1.16 + 0.670i)11-s + (0.969 − 0.316i)12-s + (−0.947 − 0.166i)13-s + (−0.242 + 0.0837i)14-s + (−0.100 − 0.0842i)15-s + (−0.960 + 0.277i)16-s + (1.00 + 0.366i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 + 0.385i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.922 + 0.385i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.19736 - 0.240042i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.19736 - 0.240042i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.927 + 1.06i)T \) |
| 19 | \( 1 + (-1.76 + 3.98i)T \) |
| good | 3 | \( 1 + (-0.306 - 1.73i)T + (-2.81 + 1.02i)T^{2} \) |
| 5 | \( 1 + (0.220 - 0.184i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (0.588 + 0.339i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (3.85 - 2.22i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (3.41 + 0.602i)T + (12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-4.15 - 1.51i)T + (13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (-0.347 + 0.413i)T + (-3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-1.03 - 2.85i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-5.24 + 9.08i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 8.82iT - 37T^{2} \) |
| 41 | \( 1 + (-1.85 + 0.326i)T + (38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-3.49 - 4.16i)T + (-7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-0.419 - 1.15i)T + (-36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (6.41 - 7.64i)T + (-9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (4.20 + 1.53i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-6.04 - 5.07i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (4.66 - 1.69i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (6.81 - 5.72i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (-0.591 - 3.35i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (1.43 + 8.15i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-8.64 - 4.99i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (10.6 + 1.87i)T + (83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (1.24 - 3.40i)T + (-74.3 - 62.3i)T^{2} \) |
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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
−14.61002226469882845360837048487, −13.21890906858892531670067107832, −12.38308413384476384359094263608, −11.01070380919014501838126462535, −10.05639303412709074110825640299, −9.439748296329264472896432670940, −7.41495058831046442074467415736, −5.43032731156648108350494198832, −4.36203839114216384318081832407, −2.88054279075999789447174420064,
2.90808517350856437628625837204, 4.99513542826094963796981883615, 6.34526311277669775861497782076, 7.59692644328172234266657256422, 8.238121818762756193502534691956, 10.05868433684998320365688616832, 11.98747985971828378191473500290, 12.54127639834369960206430509320, 13.64691105583682018704918003922, 14.28147029982022997107300215430
