| L(s) = 1 | + (−1.02 − 0.970i)2-s + (−2.92 − 0.687i)3-s + (0.116 + 1.99i)4-s + (0.533 − 4.56i)5-s + (2.33 + 3.54i)6-s + (7.85 − 3.94i)7-s + (1.81 − 2.16i)8-s + (8.05 + 4.01i)9-s + (−4.98 + 4.17i)10-s + (−13.7 − 5.91i)11-s + (1.03 − 5.91i)12-s + (−17.5 + 4.15i)13-s + (−11.9 − 3.56i)14-s + (−4.69 + 12.9i)15-s + (−3.97 + 0.464i)16-s + (−20.6 + 3.64i)17-s + ⋯ |
| L(s) = 1 | + (−0.514 − 0.485i)2-s + (−0.973 − 0.229i)3-s + (0.0290 + 0.499i)4-s + (0.106 − 0.913i)5-s + (0.389 + 0.590i)6-s + (1.12 − 0.563i)7-s + (0.227 − 0.270i)8-s + (0.894 + 0.446i)9-s + (−0.498 + 0.417i)10-s + (−1.24 − 0.538i)11-s + (0.0861 − 0.492i)12-s + (−1.34 + 0.319i)13-s + (−0.850 − 0.254i)14-s + (−0.313 + 0.864i)15-s + (−0.248 + 0.0290i)16-s + (−1.21 + 0.214i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0720i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.997 - 0.0720i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0164708 + 0.456652i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0164708 + 0.456652i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.02 + 0.970i)T \) |
| 3 | \( 1 + (2.92 + 0.687i)T \) |
| good | 5 | \( 1 + (-0.533 + 4.56i)T + (-24.3 - 5.76i)T^{2} \) |
| 7 | \( 1 + (-7.85 + 3.94i)T + (29.2 - 39.3i)T^{2} \) |
| 11 | \( 1 + (13.7 + 5.91i)T + (83.0 + 88.0i)T^{2} \) |
| 13 | \( 1 + (17.5 - 4.15i)T + (151. - 75.8i)T^{2} \) |
| 17 | \( 1 + (20.6 - 3.64i)T + (271. - 98.8i)T^{2} \) |
| 19 | \( 1 + (0.538 - 3.05i)T + (-339. - 123. i)T^{2} \) |
| 23 | \( 1 + (-13.2 + 26.4i)T + (-315. - 424. i)T^{2} \) |
| 29 | \( 1 + (15.4 - 4.62i)T + (702. - 462. i)T^{2} \) |
| 31 | \( 1 + (14.0 - 9.23i)T + (380. - 882. i)T^{2} \) |
| 37 | \( 1 + (48.4 + 17.6i)T + (1.04e3 + 879. i)T^{2} \) |
| 41 | \( 1 + (8.66 - 8.17i)T + (97.7 - 1.67e3i)T^{2} \) |
| 43 | \( 1 + (-19.9 - 26.8i)T + (-530. + 1.77e3i)T^{2} \) |
| 47 | \( 1 + (-40.0 + 60.8i)T + (-874. - 2.02e3i)T^{2} \) |
| 53 | \( 1 + (-30.4 + 17.6i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (43.5 - 18.7i)T + (2.38e3 - 2.53e3i)T^{2} \) |
| 61 | \( 1 + (0.664 - 11.4i)T + (-3.69e3 - 431. i)T^{2} \) |
| 67 | \( 1 + (-28.1 + 93.9i)T + (-3.75e3 - 2.46e3i)T^{2} \) |
| 71 | \( 1 + (66.5 + 79.2i)T + (-875. + 4.96e3i)T^{2} \) |
| 73 | \( 1 + (-25.1 - 21.0i)T + (925. + 5.24e3i)T^{2} \) |
| 79 | \( 1 + (-97.6 + 103. i)T + (-362. - 6.23e3i)T^{2} \) |
| 83 | \( 1 + (-80.6 - 76.0i)T + (400. + 6.87e3i)T^{2} \) |
| 89 | \( 1 + (85.4 - 101. i)T + (-1.37e3 - 7.80e3i)T^{2} \) |
| 97 | \( 1 + (-155. + 18.1i)T + (9.15e3 - 2.16e3i)T^{2} \) |
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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
−12.09088763369935777737126762122, −10.90512797707004554536906198343, −10.53773456583852798105912248937, −9.048347111577076202220190040434, −7.991454167657351322460902859609, −7.01136312327257395074889431536, −5.20739163863234315309401690008, −4.56666992127110043072363992110, −1.99246293965420724971762845310, −0.36115476092486400264272178336,
2.28286182914121477880994159436, 4.82639236020247432149178717985, 5.51017243337158975692913839852, 6.99895856107007015213523143591, 7.62969756058459837683474329546, 9.177845739823333146718459769419, 10.31659001232971796294142629436, 10.94342234011328704503768419083, 11.84285282276091504527904539790, 13.05002506950714440702955659656
