| L(s) = 1 | + (0.227 + 0.211i)2-s + (−0.988 − 0.149i)3-s + (−0.142 − 1.89i)4-s + (1.53 + 3.90i)5-s + (−0.193 − 0.242i)6-s + (2.06 − 1.65i)7-s + (0.755 − 0.947i)8-s + (0.955 + 0.294i)9-s + (−0.475 + 1.21i)10-s + (3.87 − 1.19i)11-s + (−0.142 + 1.89i)12-s + (−0.440 + 1.92i)13-s + (0.819 + 0.0581i)14-s + (−0.932 − 4.08i)15-s + (−3.39 + 0.511i)16-s + (−3.47 − 2.37i)17-s + ⋯ |
| L(s) = 1 | + (0.160 + 0.149i)2-s + (−0.570 − 0.0860i)3-s + (−0.0711 − 0.949i)4-s + (0.684 + 1.74i)5-s + (−0.0790 − 0.0990i)6-s + (0.779 − 0.626i)7-s + (0.267 − 0.334i)8-s + (0.318 + 0.0982i)9-s + (−0.150 + 0.382i)10-s + (1.16 − 0.360i)11-s + (−0.0410 + 0.548i)12-s + (−0.122 + 0.535i)13-s + (0.218 + 0.0155i)14-s + (−0.240 − 1.05i)15-s + (−0.848 + 0.127i)16-s + (−0.843 − 0.574i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.114i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.114i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.18688 + 0.0681745i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.18688 + 0.0681745i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.988 + 0.149i)T \) |
| 7 | \( 1 + (-2.06 + 1.65i)T \) |
| good | 2 | \( 1 + (-0.227 - 0.211i)T + (0.149 + 1.99i)T^{2} \) |
| 5 | \( 1 + (-1.53 - 3.90i)T + (-3.66 + 3.40i)T^{2} \) |
| 11 | \( 1 + (-3.87 + 1.19i)T + (9.08 - 6.19i)T^{2} \) |
| 13 | \( 1 + (0.440 - 1.92i)T + (-11.7 - 5.64i)T^{2} \) |
| 17 | \( 1 + (3.47 + 2.37i)T + (6.21 + 15.8i)T^{2} \) |
| 19 | \( 1 + (0.0899 - 0.155i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.84 - 2.61i)T + (8.40 - 21.4i)T^{2} \) |
| 29 | \( 1 + (1.17 - 0.564i)T + (18.0 - 22.6i)T^{2} \) |
| 31 | \( 1 + (-0.715 - 1.23i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.839 + 11.1i)T + (-36.5 - 5.51i)T^{2} \) |
| 41 | \( 1 + (1.93 - 2.43i)T + (-9.12 - 39.9i)T^{2} \) |
| 43 | \( 1 + (4.67 + 5.85i)T + (-9.56 + 41.9i)T^{2} \) |
| 47 | \( 1 + (1.34 + 1.24i)T + (3.51 + 46.8i)T^{2} \) |
| 53 | \( 1 + (-1.00 - 13.4i)T + (-52.4 + 7.89i)T^{2} \) |
| 59 | \( 1 + (-0.840 + 2.14i)T + (-43.2 - 40.1i)T^{2} \) |
| 61 | \( 1 + (-0.367 + 4.90i)T + (-60.3 - 9.09i)T^{2} \) |
| 67 | \( 1 + (-2.37 - 4.12i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.62 - 2.22i)T + (44.2 + 55.5i)T^{2} \) |
| 73 | \( 1 + (6.73 - 6.25i)T + (5.45 - 72.7i)T^{2} \) |
| 79 | \( 1 + (0.946 - 1.63i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (1.24 + 5.45i)T + (-74.7 + 36.0i)T^{2} \) |
| 89 | \( 1 + (-1.87 - 0.579i)T + (73.5 + 50.1i)T^{2} \) |
| 97 | \( 1 + 10.4T + 97T^{2} \) |
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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
−13.60994704344447901099006379636, −11.59087025092050789487019950394, −11.05286885273590520825413981203, −10.26270072771267058948109246885, −9.295598035048635931159891924886, −7.20459519643554768395670215440, −6.60132300351927982169982775034, −5.63119515256350586470458608837, −4.08123445807310160210649802371, −1.87918058371259309028394978575,
1.80645741536240899905300562209, 4.28009504150264367117910604857, 5.01998542803296268230342252785, 6.32776178160455495059894684140, 8.158557417182511315205960187673, 8.751609494188992943919867807761, 9.815861576346793446775357750092, 11.48965397616077541628260718270, 12.12229606402540253016496176270, 12.81899983825930712979605254511
