| L(s) = 1 | + (2.60 − 0.195i)2-s + (−1.58 + 0.691i)3-s + (4.75 − 0.716i)4-s + (−0.564 − 2.47i)5-s + (−3.99 + 2.10i)6-s + (1.46 − 2.20i)7-s + (7.14 − 1.63i)8-s + (2.04 − 2.19i)9-s + (−1.95 − 6.33i)10-s + (−0.836 + 1.73i)11-s + (−7.05 + 4.42i)12-s + (−6.25 + 0.468i)13-s + (3.37 − 6.02i)14-s + (2.60 + 3.54i)15-s + (9.09 − 2.80i)16-s + (4.89 + 0.737i)17-s + ⋯ |
| L(s) = 1 | + (1.84 − 0.137i)2-s + (−0.916 + 0.399i)3-s + (2.37 − 0.358i)4-s + (−0.252 − 1.10i)5-s + (−1.63 + 0.860i)6-s + (0.552 − 0.833i)7-s + (2.52 − 0.576i)8-s + (0.681 − 0.731i)9-s + (−0.617 − 2.00i)10-s + (−0.252 + 0.523i)11-s + (−2.03 + 1.27i)12-s + (−1.73 + 0.130i)13-s + (0.901 − 1.61i)14-s + (0.673 + 0.914i)15-s + (2.27 − 0.701i)16-s + (1.18 + 0.178i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.775 + 0.631i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.775 + 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.87366 - 1.02254i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.87366 - 1.02254i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.58 - 0.691i)T \) |
| 7 | \( 1 + (-1.46 + 2.20i)T \) |
| good | 2 | \( 1 + (-2.60 + 0.195i)T + (1.97 - 0.298i)T^{2} \) |
| 5 | \( 1 + (0.564 + 2.47i)T + (-4.50 + 2.16i)T^{2} \) |
| 11 | \( 1 + (0.836 - 1.73i)T + (-6.85 - 8.60i)T^{2} \) |
| 13 | \( 1 + (6.25 - 0.468i)T + (12.8 - 1.93i)T^{2} \) |
| 17 | \( 1 + (-4.89 - 0.737i)T + (16.2 + 5.01i)T^{2} \) |
| 19 | \( 1 + (-6.22 - 3.59i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.42 - 1.93i)T + (5.11 - 22.4i)T^{2} \) |
| 29 | \( 1 + (4.50 - 1.76i)T + (21.2 - 19.7i)T^{2} \) |
| 31 | \( 1 + (-1.58 - 0.912i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.637 - 1.62i)T + (-27.1 + 25.1i)T^{2} \) |
| 41 | \( 1 + (5.19 - 4.82i)T + (3.06 - 40.8i)T^{2} \) |
| 43 | \( 1 + (-2.51 - 2.33i)T + (3.21 + 42.8i)T^{2} \) |
| 47 | \( 1 + (0.676 + 9.02i)T + (-46.4 + 7.00i)T^{2} \) |
| 53 | \( 1 + (2.80 + 1.10i)T + (38.8 + 36.0i)T^{2} \) |
| 59 | \( 1 + (1.03 + 0.960i)T + (4.40 + 58.8i)T^{2} \) |
| 61 | \( 1 + (1.94 - 12.9i)T + (-58.2 - 17.9i)T^{2} \) |
| 67 | \( 1 + (-1.52 + 2.64i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (9.12 - 7.27i)T + (15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (-3.85 + 5.65i)T + (-26.6 - 67.9i)T^{2} \) |
| 79 | \( 1 + (2.00 + 3.46i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.241 + 3.22i)T + (-82.0 - 12.3i)T^{2} \) |
| 89 | \( 1 + (0.423 - 5.64i)T + (-88.0 - 13.2i)T^{2} \) |
| 97 | \( 1 + (-2.32 - 1.33i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.63345566019766838520065794942, −10.31454641055224538397686183483, −9.777885639104762577196247796218, −7.71511179504434664303665223354, −7.14074308645164943215636410491, −5.66808638388906206847110097694, −5.03856672580131909860136786338, −4.51639755359090715058119032964, −3.48420815643836552053074988496, −1.46951496537667571783106854504,
2.32423754796252648324033649974, 3.20194563308757249087752182017, 4.79611730384126262953790084267, 5.40137856091978624579077105840, 6.18936550490078928152650347797, 7.35264405734219141836780298001, 7.60758675394527378925031428469, 9.865590480588531746825980999651, 10.95426867675387070359723332975, 11.54685827617927973059837801589
