| 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
−11.54685827617927973059837801589, −10.95426867675387070359723332975, −9.865590480588531746825980999651, −7.60758675394527378925031428469, −7.35264405734219141836780298001, −6.18936550490078928152650347797, −5.40137856091978624579077105840, −4.79611730384126262953790084267, −3.20194563308757249087752182017, −2.32423754796252648324033649974,
1.46951496537667571783106854504, 3.48420815643836552053074988496, 4.51639755359090715058119032964, 5.03856672580131909860136786338, 5.66808638388906206847110097694, 7.14074308645164943215636410491, 7.71511179504434664303665223354, 9.777885639104762577196247796218, 10.31454641055224538397686183483, 11.63345566019766838520065794942
